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The Role of Stochastic Forcing in Generating ENSO Diversity
ERIN E. THOMAS AND DANIEL J. VIMONT
Department of Atmospheric and Oceanic Sciences, and Nelson Institute Center for Climatic Research, University of
Wisconsin–Madison, Madison, Wisconsin
MATTHEW NEWMAN
CIRES, University of Colorado Boulder, and NOAA/Earth System Research Laboratory/Physical Sciences Division,
Boulder, Colorado
CÉCILE PENLAND
NOAA/Earth System Research Laboratory/Physical Sciences Division, Boulder, Colorado
CRISTIAN MARTÍNEZ-VILLALOBOS
Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California
(Manuscript received 28 August 2017, in final form 28 August 2018)
ABSTRACT
Numerous oceanic and atmospheric phenomena influence El Niño–Southern Oscillation (ENSO) variability, complicating both prediction and analysis of the mechanisms responsible for generating ENSO diversity. Predictability of ENSO events depends on the characteristics of both the forecast initial conditions
and the stochastic forcing that occurs subsequent to forecast initialization. Within a linear inverse model
framework, stochastic forcing reduces ENSO predictability when it excites unpredictable growth or interference after the forecast is initialized, but also enhances ENSO predictability when it excites optimal
initial conditions that maximize deterministic ENSO growth. Linear inverse modeling (LIM) allows for
straightforward separation between predictable signal and unpredictable noise and so can diagnose its own
skill. While previous LIM studies of ENSO focused on deterministic dynamics, here we explore how noise
forcing influences ENSO diversity and predictability. This study identifies stochastic forcing details potentially contributing to the development of central Pacific (CP) or eastern Pacific (EP) ENSO characteristics.
The technique is then used to diagnose the relative roles of initial conditions and noise forcing throughout the
evolution of several ENSO events. LIM results show varying roles of noise forcing for any given event, highlighting its utility in separating deterministic from noise-forced contributions to the evolution of individual
ENSO events. For example, the strong 1982 event was considerably more influenced by noise forcing late in its
evolution than the strong 1997 event, which was more predictable with long lead times due to its deterministic
growth. Furthermore, the 2014 deterministic trajectory suggests that a strong event in 2014 was unlikely.
1. Introduction
El Niño–Southern Oscillation (ENSO) is the dominant
source of interannual climate variability on Earth (see,
e.g., Wallace et al. 1998), with large impacts on global
Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/
JCLI-D-17-0582.s1.
Corresponding author: Erin Thomas, eethomas3@wisc.edu
climate patterns (Diaz et al. 2001; Alexander et al. 2002).
While theories exist to explain the gross spatial and temporal characteristics of ENSO event evolution (Zebiak
and Cane 1987; Battisti 1988; Suarez and Schopf 1988;
Battisti and Hirst 1989; Penland and Sardeshmukh 1995;
Jin 1997; Neelin et al. 1998), individual ENSO events show
considerable diversity in their spatial details, evolution,
predictability, and timing [see Capotondi et al. (2015) for a
review]. Spatial diversity, often characterized using terms
such as central Pacific (CP) or eastern Pacific (EP) (e.g.,
Ashok et al. 2007; Kao and Yu 2009; Kug et al. 2009, 2010a;
DOI: 10.1175/JCLI-D-17-0582.1
Ó 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).
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Yu and Kim 2011; Newman et al. 2011b; Kim et al. 2012a;
Kim et al. 2012b; Capotondi 2013; Capotondi et al. 2015),
arises due to the varying roles of different physical processes allowing any given ENSO event to have a continuous range of CP and EP characteristics (Karnauskas 2013;
Capotondi et al. 2015). This study provides a general
framework to identify and characterize the role of these
physical processes in generating ENSO diversity within the
context of the dynamically evolving ENSO system.
Several studies have shown that atmospheric and oceanic midlatitude variability help generate initial structures in the tropical Pacific that lead to ENSO diversity.
Vimont et al. (2014) use linear inverse modeling (LIM;
Penland and Sardeshmukh 1995) to identify optimal initial conditions leading to ENSO events with either CP or
EP characteristics. They find that CP-related initial conditions are associated with the Pacific meridional mode
(PMM; Chiang and Vimont 2004) through the ‘‘seasonal
footprinting mechanism’’ (Vimont et al. 2001, 2003a,
2003b, 2009; Zhang et al. 2009; Yu and Kim 2011; Park
et al. 2013). Southern Hemisphere atmospheric variability has also been shown to generate initial heat content
anomalies that can influence EP-type ENSO events (You
and Furtado 2017). Other studies show that ENSO diversity is associated with initial oceanic heat content
anomalies, ocean dynamics, and basin-wide thermocline
variations (Meinen and McPhaden 2000; Kao and Yu
2009; Kug et al. 2010a; Horii et al. 2012; Fedorov et al.
2015). Finally, ENSO initial conditions have been associated with forcing from the tropical Indian Ocean or from
the subtropical Atlantic (e.g., Penland and Matrosova
2006, 2008; Rodríguez-Fonseca et al. 2009; Martín-Rey
et al. 2012).
Variations in stochastic forcing (here we loosely define
stochastic or external forcing as forcing that is external to
the essential deterministic dynamics that produce ENSO
variability) have also been shown to generate ENSO diversity. Newman et al. (2011b) show that ENSO diversity
arises naturally in a stochastically forced linear inverse
model simulation with an unchanging dynamical description, confirming that stochastic forcing is a sufficient
condition for generating ENSO diversity. Studies also
show that ENSO events with strong EP characteristics,
such as the 1997/98 El Niño event, are influenced by
westerly wind busts (WWBs) and zonal wind variability in
the western tropical Pacific (Boulanger and Menkes 1999;
McPhaden 1999; Harrison and Chiodi 2009; Kug et al.
2010b; Fedorov et al. 2015; Chen et al. 2015). It has also
been shown that strong ENSO events, which typically
have EP characteristics, are more strongly influenced by
state-dependent wind stress variations (Perez et al. 2005;
Gebbie et al. 2007; Kapur and Zhang 2012; Levine
et al. 2016).
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LIM provides an observationally based, empirical
model that approximates slow dynamical processes via a
deterministic linear dynamical operator, and the effect
of fast processes as Gaussian white noise. Penland and
Sardeshmukh (1995) use LIM to show that ENSO
growth can occur via nonnormal processes in a linearly
stable dynamical system. In that case, stochastic forcing
both maintains climatological variance and explains the
observed temporal irregularities of ENSO variability.
Penland (1996) shows that a time-independent linear
approximation of the slow dynamics of the Indo-Pacific
forced by seasonally varying noise can account for observed temporal ENSO behaviors, such as the seasonal
ENSO phase locking.
Demonstrated forecast skill of the LIM (e.g., Penland
and Magorian 1993; Penland and Matrosova 1998;
Newman 2007; Alexander et al. 2008; Newman et al.
2011a; Newman and Sardeshmukh 2017) motivated its
use as a diagnostic tool for understanding the physical
processes that generate ENSO variability. LIM allows
for an objective, observationally grounded calculation
of the initial state that maximizes deterministic growth
toward a specified final state. Previous studies show that
the optimal initial conditions maximizing tropical Pacific
SST growth contain SST anomalies north of the equator
and in the far eastern equatorial Pacific (Penland and
Sardeshmukh 1995; Newman et al. 2011a). Newman
et al. (2011a) also show that the optimal initial conditions contain positive thermocline depth anomalies
throughout the central tropical Pacific. Capotondi and
Sardeshmukh (2015) use LIM to show the importance of
the thermocline initial state and suggest SST precursors
alone are not sufficient to capture the development of
ENSO diversity.
A variety of physical processes have been proposed to
explain ENSO diversity. In this study, we take a forecast
perspective in which the evolution of an individual
ENSO event from a given time depends on (i) the specific set of initial conditions that determine how the
system will evolve along its deterministic trajectory, as
well as (ii) the ‘‘external’’ forcing that, when convolved
with the dynamics, pushes the system away from (in
general) the deterministic trajectory. A third possibility
exists that ENSO diversity is caused by variations in the
internal dynamics responsible for ENSO evolution (e.g.,
Fedorov and Philander 2000, 2001; An and Wang 2000;
An and Jin 2000; Wang and An 2002; Yeh et al. 2009;
Capotondi and Sardeshmukh 2017). Here, we focus
solely on the role of the initial conditions and stochastic
forcing in generating ENSO diversity.
Although previous studies have characterized the optimal initial conditions for maximum CP and EP growth
(Vimont et al. 2014; Capotondi and Sardeshmukh 2015),
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the specific structures within the stochastic forcing that
generate these initial conditions have not been identified.
In this study, we characterize the role of stochastic forcing
mechanisms in generating CP or EP characteristics using
a LIM forecast framework to separate the deterministic
evolution of the tropical Pacific from the unpredictable, or
noise forced, evolution. Within the forecasting framework,
it is possible the noise forcing can excite optimal initial
conditions that enhance deterministic growth and, thus,
improve the predictability of an event. However, the same
stochastic forcing may reduce the predictability of the
event if it excites the growth of prediction errors after the
forecast is initialized or perturbs the state of the tropical
Pacific away from expected ENSO growth.
The analysis herein is strongly motivated by the results
of Newman and Sardeshmukh (2017), who show that
forecasts of tropical Indo-Pacific SST anomalies using
LIM have comparable skill to multimodel ensemble
mean forecasts made using the full nonlinear coupled
ocean–atmosphere models of the North American
Multimodel Ensemble (NMME; Kirtman et al. 2014).
Both forecast techniques have spatial and temporal
variations of skill that are similar both to each other and
to the potential skill estimated from the forecast signalto-noise ratios within a perfect linear inverse model
framework. This suggests that the deterministic evolution of ENSO evolution can be estimated by a linear
approximation to the dynamics. As such, the linear inverse model can be used as a ‘‘filter’’ to remove the
deterministic component and, hence, to diagnose the
role of noise in the evolution of individual ENSO events;
this is not easily accomplished using the full nonlinear
GCMs. Following the methods outlined in Penland
and Hartten (2014), we empirically calculate the noise
forcing conducive to CP and EP growth directly from
observations. We then identify the physical mechanisms
within the noise forcing that lead to CP or EP growth
and estimate the role of noise forcing during the 1982/83
and 1997/98 EP ENSO events, the 2009/10 CP ENSO
event, the 2015/16 ENSO event, and the ‘‘failed’’ event
of 2014. The implications for CP and EP ENSO predictability will be discussed.
This paper is organized as follows. Section 2 of this
paper describes the methods for empirically estimating
the linear dynamics of the tropical Pacific from observations, the optimal initial conditions that maximize CP
and EP growth. Section 3 explains the methodology for
calculating the noise forcing of the CP and EP optimal
initial conditions and presents the structures within the
noise forcing that lead to CP and EP growth. Section 4
analyzes the role of the noise forcing and dynamics
during past CP and EP ENSO events. Section 5 discusses
the implications of the results.
2. Methods
a. Data
The linear inverse model is developed using monthly
optimally interpolated SST (OISST; Reynolds et al.
2002) in the tropical Pacific (258S–258N, 1208–2858E)
and monthly thermocline depth, calculated as the depth
of the 208 isotherm (Z20) from the NCEP Global Ocean
Data Assimilation System (GODAS; Behringer and
Xue 2004), between 258S–258N and 1208–2858E. The
SST and Z20 data, from 1982 to 2016, are averaged onto
28 latitude 3 58 longitude grids. The annual cycle for each
dataset is removed by subtracting the 1982–2016 monthly
climatological mean from each month. The monthly
anomalies are then smoothed with a 3-month running
mean and detrended using linear regression. We note that
Vimont et al. (2014) develop a linear inverse model using
the HadISST product (Rayner et al. 2003) and SODA
reanalysis (Carton and Giese 2008), and recover the same
results as shown in Fig. 1.
EOF analysis is applied as a prefilter to the monthly
SST and Z20 anomalies, so that our analysis is done in
the space of the leading nine SST EOFs and three Z20
EOFs. For reference, the leading two EOFs of SST and
Z20 are shown in Figs. S1a–d in the online supplemental
material. The leading EOF of SST (explaining 52.1% of
the variance) shows a typical canonical ENSO pattern,
while the second EOF (12.1% of the variance) strongly
projects onto the SST pattern known as the PMM. For
reference, the temporal correlation between the PMM
time series and PC2 is r 5 0.82. Supplemental Figs. S1e
and S1f show the SST patterns of CP and EP events
based upon the Takahashi et al. (2011) definition for the
C and E indices as follows:
PC1 PC2
C 5 pffiffiffiffiffi 1 pffiffiffiffiffi
l1
l2
!,
pffiffiffi
2
!,
pffiffiffi
PC1 PC2
2,
E 5 pffiffiffiffiffi 2 pffiffiffiffiffi
l1
l2
(1)
where PC1 is the first principal component of SST, PC2
is the second principal component, and l1 and l2 are the
corresponding eigenvalues. These definitions are used in
section 2b to specify the direction of growth in order to
estimate the CP and EP optimal initial conditions.
SST pentad means are calculated from daily OISST
data from 1982 to 2016. The Z20 pentad means are calculated from daily GODAS thermocline depths from
1982 to 2016 (again, the thermocline depth is calculated as
the depth of the 208 isotherm). Both SST and Z20 pentad
data are averaged onto the same 28 latitude 3 58 longitude
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FIG. 1. SST (8C; shading) and thermocline depth (m; black contours) for t 5 6 month CP and EP (a),(c) optimal
initial conditions and (b),(d) final states. SST contour interval is 0.48C for the optimals and 0.88C for the final
conditions. Thermocline depth contour interval is 4 m for the optimals and 8 m for the final conditions. Positive
(negative) thermocline depth anomalies correspond to solid (dashed) contours. The zero contour has been omitted.
grid as the monthly data. The annual cycle is removed,
the data detrended and smoothed with a five-pentad
running mean (see section 3b). Similarly, pentad means
of several different global quantities are calculated from
daily NCEP–NCAR Reanalysis 1 data (Kalnay et al.
1996) between 1982 and 2016. To determine anomalies,
the annual cycle is removed and the data detrended for
each variable.
b. Linear inverse model
LIM approximates the evolution of a dynamical system,
in this case the tropical Pacific, by a multivariate linear
model as follows (Penland and Sardeshmukh 1995):
dx
5 Lx 1 j,
dt
(2)
where x is the anomalous state of the system, L is the
dynamical system matrix representing the linearized
approximation to the dynamics of the system (including
the linear approximation to the nonlinear dynamics),
and j is the white noise forcing. The dynamical system
matrix L can be empirically estimated from a set of observations as presented below, and an empirical estimation of the noise forcing j is presented in section 2b.
Previous studies show the tropical Pacific is well represented by this stochastically forced linear system (Penland
and Sardeshmukh 1995; Penland 1996). Within this linear
inverse model framework, the evolution of the state x can
be described as
x(t) 5 eL(t2t0 ) x(t0 ) 1
ðt
eL(t2s) j(s) ds,
(3)
t0
where the first term on the right-hand side of the equation
represents the deterministic, or predicable, evolution of the
system from time t0 to t (see, e.g., Chang et al. 2004) and the
second term represents the nondeterministic, unpredictable, or noise forced, part of the system. Note that the
second term convolves the stochastic forcing in (2) with the
deterministic dynamical evolution through the remaining
forecast time. If the linear model perfectly represents the
system, the second term is equal to the forecast error.
It can be shown that the correlation between x(0) and
the second term of (3) is zero. We define Gt the ‘‘Green
function’’ or propagator matrix as the matrix yielding
the linear inverse model forecast x0 (t) when operating
on initial conditions x(0):
x0 (t) 5 eLt x(0) [ Gt x(0),
(4)
where x0 (t) is the forecast of the final state at time t and
x(0) is the initial state. From (4), one can derive Gt and
L from the lagged covariance statistics of the system:
Gt 5 Ct C21
0
L 5 ln(Gt )/t,
and
(5)
(6)
where Ct is the t-lag covariance matrix of the state
vector x and C0 is the zero-lag covariance matrix of the
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state vector x. As previously mentioned, the linear operator L governs the deterministic evolution of the system
over a specific time period t. The maximum amplification
(e.g., Penland and Sardeshmukh 1995, Fig. 4) curve indicates that the system experiences maximum growth
over a time period of about 9 months. The growth curve
of the error energy intersects the maximum amplification
curve around 23 months, which represents the predictability limit of the system (results not shown).
We define the state vector x as in Vimont et al. (2014)
with sea surface temperature (SST) and thermocline
depth (Z20) as follows:
x 5 (ZSST , ZZ20 ),
(7)
where ZSST represents the leading nine SST principal
components (PCs) (88.1% of the variance) and ZZ20 the
leading three Z20 PCs (45.8% of the variance). The PCs
of the state vector are normalized by the square root of
the total variance of the respective field; that is, ZSST is
normalized by the square root of the sum of all SST eigenvalues, and ZZ20 is normalized by the square root of
the sum of all Z20 eigenvalues.
c. Optimal initial conditions
The noise forcing that will maximize the deterministic
growth toward CP or EP events will be the noise that
pushes the system toward generating the associated
optimal initial conditions. Therefore, we first calculate
the optimal initial conditions that maximize either CP
or EP growth. Using the previously calculated linear
dynamics of the system (6), we are able to determine
the initial state that maximizes the deterministic growth
in a given direction (Penland and Sardeshmukh 1995;
Newman et al. 2011a; Vimont et al. 2014). The resulting
initial structures are, hereafter, referred to as optimal
initial conditions. The optimal initial conditions that
maximize growth toward the CP and EP final conditions
are calculated by following the methodology of Vimont
et al. (2014). Based upon the definition of CP and EP
events (1), we define the vector coordinate directions for
the CP and EP events as follows:
(
1
1
nCP 5 pffiffiffiffiffiffiffi , pffiffiffiffiffiffiffi , 0, 0, . . .
2l1
2l2
(
)
)
1
1
nEP 5 pffiffiffiffiffiffiffi , 2pffiffiffiffiffiffiffi , 0, 0, . . . ,
2l1
2l2
(8)
where l1 and l2 are the leading two SST eigenvalues.
The CP and EP final norm kernels (N), which define the
direction of growth, are defined as
NCP 5 nTCP nCP 1 I
NEP 5 nTEP nEP 1 I ,
(9)
where I is the identity matrix times some arbitrary small
number ( 5 1029 ) that is necessary for numerical stability (Tziperman et al. 2008). Using the CP and EP
norms above, we estimate the optimal initial conditions
p that maximize growth m in the direction of the chosen
norm N over a finite time period t by solving the generalized eigenvalue problem:
GTt NGt p 2 m(t)p 5 0.
(10)
Figures 1a and 1c show the optimal initial conditions
that maximize either CP or EP growth, respectively,
over a 6-month time period. Results are similar for
t 5 3 months and t 5 9 months (not shown). The final
CP and EP ENSO states are shown in Figs. 1b and 1d,
respectively. Optimal initial and associated final conditions found here are very similar to those previously
calculated in Vimont et al. (2014); we show the structures here since the linear inverse model in this study is
constructed with different SST and thermocline datasets.
The optimal initial conditions maximizing CP growth
show a spatial SST structure representative of the PMM,
along with an increased zonal thermocline gradient with
deep thermocline anomalies in the western Pacific and
shoaled thermocline anomalies in the eastern Pacific. The
EP optimal initial conditions contain a zonal SST gradient containing positive SST anomalies located in the far
eastern Pacific and negative SST anomalies in the western
and central Pacific. The EP optimal conditions also suggest that a deepened thermocline across much of the
central equatorial Pacific maximizes EP growth. The
optimal structures are insensitive to the number of EOFs
retained in the state vector, which is consistent with the
results from Vimont et al. (2014).
The temporal evolution patterns of the L2 (i.e., the
Euclidean norm), CP, and EP optimal initial conditions
(red lines) together with SST PC1 and the CP and EP
indices (black lines) are shown in Figs. 2a–c. The time
series of the optimal initial conditions are calculated by
projecting the state vector x (7) onto the L2, CP, or EP
optimal initial structures. As expected, the L2, CP, and
EP optimal initial time series lead the SST PC1, CP index, and EP index, respectively. The maximum lagged
correlation for both CP and EP occurs when the optimal
leads the index by 2 months.
3. Noise forcing of CP/EP optimals
This section describes the methods used to estimate
the noise forcing that drives the tropical Pacific toward a
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FIG. 2. The temporal evolution of (a) SST PC1 (black) and L2 optimal initial conditions (red), (b) CP index
(black) and CP optimal initial conditions (red), and (c) EP index (black) and EP optimal initial conditions (red).
The noise forcing time series associated with 6-month (d) CP and (e) EP optimal initial conditions are shown. The
CP and EP noise forcing time series in (d) and (e) are calculated as the projection of j(t) [see (11)] onto the CP or
EP optimal initial conditions (Fig. 1). (f) The seasonal variance of the total noise forcing j(t) is shown as is (g) the
seasonal variance of the noise forcing associated with the L2 (blue), CP (red), and EP (black) optimals. The variance is smoothed with successive five- and seven-pentad running means. The power spectra (solid black) of the
unfiltered (h) CP and (i) EP noise forcing time series. The solid red lines show the white noise null hypothesis, while
the dashed red lines show the 95% confidence interval.
particular state, especially the optimal initial conditions for CP or EP ENSO events. We then diagnose
specific physical mechanisms that contribute to that
noise forcing.
a. Stochastic forcing
The observed stochastic forcing, or noise forcing, of
the system [see (2)] can be estimated using a centereddifference approximation to (2) following the methodology described in Penland and Hartten (2014):
j(t) ’
[x(t 1 Dt) 2 x(t 2 Dt)]
2 Lx,
2Dt
(11)
where x is now the finely resolved (in time) state vector
based on pentad data and Lx is the deterministic evolution. Note here that the right-hand side of (11) can be
interpreted as a ‘‘dynamical filter’’ that removes the
deterministic tendency (Lx) from the actual tendency.
The result (left-hand side) is an estimate of the broadband forcing that the slow dynamics will see as white at
the frequencies of interest to El Niño (see Figs. 2h,i).
The resulting multivariate time series j(t) empirically
estimates the nondeterministic component of the system’s tendency (i.e., the noise forcing) and not the deterministic evolution of ENSO itself. This is critical to
ensuring that the resulting noise patterns are not simply
aliasing the deterministic component of ENSO’s evolution. In other words, because (i) the linear inverse
model largely reproduces both seasonal and year-toyear variations of tropical Indo-Pacific SST forecast skill
from fully nonlinear models (see Figs. 2 and 3 in
Newman and Sardeshmukh 2017) and (ii) the linear
inverse model describes the deterministic evolution of
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THOMAS ET AL.
the system, we can use (11) to filter out the deterministic
component, thereby retaining the nondeterministic contribution from the noise.
Note that although we have calculated L from monthly
(i.e., coarse grained) data, in (11) we estimate the noise
from pentad (i.e., more finely resolved) data. The reason
for this distinction is described in the cautionary note
from Penland and Hartten (2014; see the supplemental
material therein) on the centered difference approximation to the continuous Stratonovich system in (2). In a
Stratonovich system the contemporaneous correlation
between the noise and the system state is nonzero, and
the centered-difference equation approximation in (11) is
only valid in the limit as Dt goes to zero. In this case, the
approximation is justified if the time scale over which (11)
is evaluated is short compared to that of the deterministic
evolution of the system. Hence, we evaluate (11) using
pentad data. Additionally, we have confirmed that the
noise covariance matrix estimated from (11) is consistent
with the noise covariance matrix calculated using the
fluctuation–dissipation relationship with stationary statistics (Penland and Sardeshmukh 1995; for further details see section S2a in the online supplemental material).
The finely resolved state vector x is defined as the
projection of the pentad SST and Z20 anomalies onto
the leading nine SST and three Z20 EOFs that were
identified from monthly data (see section 2a). As before,
these time series (nine SST and three Z20) of the
finely resolved state vector x are normalized by the
square root of the total variance of each variable’s
monthly anomalies. Finally, the pentad-evolving multivariate noise is estimated as a residual from (11), resulting in a multivariate dataset with 12 degrees of
freedom (nine SST and three Z20) and 2555 pentads
(73 3 35 yr of data).
b. Noise structures associated with CP and EP
optimals
The noise time series associated with forcing any specified optimal initial conditions is estimated by projecting
the total noise forcing j(t) in (11) onto that optimal’s
spatial structure. Figures 2d and 2e show the resulting time
series of noise forcing associated with the CP and EP
6-month optimals, respectively. A Kolmogorov–Smirnov
test applied to both noise time series shows their distributions are not significantly non-Gaussian (section S2c
in the supplemental material). The linear inverse model
assumes that relative to its deterministic time scales, the
noise forcing will be spectrally white, which is confirmed
by the spectral analysis of the raw noise in Figs. 2h and 2i.
Further details regarding the evaluation of the spectral
behavior of the noise are provided in the supplemental
material (section S2b). The decrease in power at high
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frequencies is expected due to the centered-differencing
method used to estimate the noise. Due to this characteristic, we apply a five-pentad running mean to the SST and
Z20 data prior to calculating the noise time series to remove
the insignificant high-frequency variance in the noise.
Several previous studies also show the low-frequency part
of the noise spectrum is more effective in forcing ENSO
(Roulston and Neelin 2000; Newman et al. 2009; Levine
and Jin 2010; Lopez et al. 2013).
Figure 2f shows the seasonal variation of the total
noise forcing amplitude, which reaches a maximum
(minimum) during March (August). Figure 2g shows the
seasonal variation of the noise amplitudes associated
with the L2 (blue), CP (red), and EP (black) optimals.
The variance was smoothed with successive five- and
seven-pentad centered running means. CP noise amplitudes have the weakest seasonality but are highest during boreal winter (DJF). The EP noise variance, on the
other hand, contains much stronger seasonality and is
highest during boreal spring (MAM) and lowest during
fall (SON). Unlike Penland (1996), there is no semiannual seasonal cycle in the noise variance, likely because of the inclusion of subsurface data in this analysis.
To identify physical mechanisms within the stochastic
forcing contributing to CP and EP optimal initial condition development, we regress the global pentad-averaged
anomalies onto the stochastic forcing time series in
Figs. 2d and 2e. The resulting regression maps, shown
for each season in Figs. 3 and 4, may be interpreted as
variability in a particular field that covaries with the
noise forcing of a given set of optimal initial conditions
and are robust to modifications of the number of EOFs
retained in the state vector. While this technique does
not necessarily identify specific mechanisms that
guarantee ENSO growth, it does identify atmospheric
patterns that covary with the noise forcing of the CP
and EP optimals. These patterns are consistent with a
number of mechanisms previously proposed in the literature, as discussed below. See the supplemental material for full regression maps and a discussion of the
statistical significance.
Figure 3 shows sea level pressure (SLP), 850-mb wind,
and outgoing longwave radiation (OLR) anomalies regressed on the noise forcing time series associated with
the 6-month CP (left column) and EP (right column)
optimal initial conditions. Figure 4 shows the corresponding regression patterns of ocean–atmosphere thermal flux (OAFLUX, defined as sensible heat flux 1 latent
heat flux; positive upward) and surface wind stress
anomalies. Note that the maps in Figs. 3 and 4 contain a
large range of variabilities, which is not surprising
considering that ‘‘noise forcing’’ could include a variety
of phenomena.
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FIG. 3. Atmospheric noise structures associated with (left) CP and (right) EP optimal
initial conditions. Shown are the seasonal regression coefficients between sea level pressure
(SLP; hPa; contours), 850-mb wind (m s21; vectors), and OLR (W m22; shading), as well as
the noise forcing time series of the CP or EP optimal initial conditions. The first row shows
the regression coefficients of boreal winter (DJF). The second through fourth rows show the
regression coefficients for the boreal spring (MAM), summer (JJA), and fall (SON) months,
respectively. Positive (negative) SLP is indicated with red (blue) contours where the contour
interval is every 0.5 hPa. The zero contour has been omitted. OLR is defined as positive
upward. Wind vectors are only shown where the geometric sum of the correlation coefficients
is equal to or greater than 0.1.
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FIG. 4. Noise-forced oceanic structures associated with (left) CP and (right) EP optimal initial conditions. Shown are the seasonal regression coefficients between the
surface wind stress (N m22; vectors) and ocean–atmosphere flux (OAFLUX; W m22;
shading) and the noise forcing time series of the CP or EP optimal initial conditions. The
first row shows the regression coefficients of boreal winter (DJF). The second through
fourth rows show the regression coefficients for the boreal spring (MAM), summer
(JJA), and fall (SON) months, respectively. OAFLUX is defined as positive upward.
Wind stress vectors only shown where the geometric sum of the correlation coefficients is
equal to or greater than 0.1.
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The CP SLP structures (Fig. 3, left) show typically
higher noise amplitudes during winter and spring in each
hemisphere. In the Northern Hemisphere, the CP SLP
forcing structures during boreal fall (SON), winter
(DJF), and spring (MAM) are similar to those of the
North Pacific Oscillation (NPO). During these seasons,
the OAFLUX (Fig. 4, left) shows a flux into the ocean
with a pattern corresponding to the PMM. These results
support previous studies that show the NPO and PMM
are related to initiating CP ENSO events (Yu and Kim
2011; Kim et al. 2012b). In the Southern Hemisphere,
subtropical SLP anomalies contribute to CP noise
forcing throughout the year, associated with only weak
downward heat flux anomalies.
For forcing the EP optimal, on the other hand, the
Northern Hemisphere SLP noise is much less important
except perhaps during winter, whereas the Southern
Hemisphere SLP noise appears important throughout
the year (Fig. 3, right). In particular, a dipolar SLP
structure is evident during austral fall (MAM) and
spring (SON) that resembles the ‘‘South Pacific Oscillation,’’ which has been identified as an EP ENSO precursor by You and Furtado (2017). Similar SLP patterns
are also found in Penland and Matrosova (2008). The
850-mb wind (Fig. 3, right) and surface wind stress
(Fig. 4, right) patterns show positive zonal wind anomalies in the equatorial Pacific throughout the year,
although there are slight seasonal differences in the location and magnitude. The zonal winds lie south of the
equator in the western tropical Pacific during DJF,
centered close to the equator during MAM and JJA, and
north of the equator in the western tropical Pacific
during SON. The wind anomalies are weakest during boreal summer (JJA). Since the EP noise variance
peaks in boreal spring (MAM; Fig. 2f), the springtime noise structures are of particular interest. The
springtime (MAM) stochastic forcing of EP optimals
shows strong low-level zonal wind anomalies in the
western and central tropical Pacific (Fig. 3, right).
Interestingly, the ocean–atmosphere flux shows a flux
into the ocean in the Southern Hemisphere off the
South American coast from DJF through JJA (Fig. 4,
right), suggestive of the South Pacific meridional
mode (SPMM).
4. Role of noise forcing versus deterministic
dynamics during past ENSO events
In the previous section, we identified noise structures
related to the excitation of an optimal set of initial conditions, which will lead to maximum deterministic growth
[first term on the right-hand side of (3)] over some time
period. In this section, over the lifetime of different
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ENSO events, we compare the deterministic evolution
from initial conditions to the nondeterministic evolution
driven by noise. We note that the nondeterministic
evolution of the system is controlled by the second term
in (3). Chang et al. (2004) show that the forcing structure
optimizing error growth over a finite time period (time
0 to t) is the leading eigenvector of B(t), where
B(t) 5
ðt
eL*(t2s) eL(t2s) ds.
(12)
0
It is worth noting the relationship between the initial
optimals investigated in section 3 and the noise structures
that dominate the nondeterministic trajectory. At a given
time t 0 after the forecast initialization, but before a
forecast target end time t, the relationship between the
initial optimals and the optimal noise structure can be
illuminated by writing (12) as an infinite sum:
B(t) 5 lim
n/‘
n
å
k51
eL*(t2kds) eL(t2kds) ds
(13)
(where ds 5 t/n). Note that at time t 0 the argument inside the sum is simply GTt2t0 Gt2t0 . Hence, the structure
that optimally perturbs a system away from its trajectory
at time t 0 between 0 and t is simply the optimal set of
initial conditions at time t 2 t 0 . That is, at some time t 0
after the forecast initialization, the noise structure that
optimally ruins a forecast is that which experiences the
most transient growth over the remaining forecast time
(t 2 t0 ). This argument can easily be generalized to investigate growth toward or away from specific norms,
and is used to investigate the role of noise forcing in the
generation of ENSO events with specific CP or EP
characteristics herein. We use the 6-month optimal initial structure for analysis herein; results are not sensitive
to the choice of optimization time.
To highlight how the role of the noise forcing can vary
for any given event, we select two strong EP events
(1982/83 and 1997/98), one strong CP event (2009/10),
the recent 2015/16 event (which displays both CP and
EP characteristics), and the ‘‘failed’’ 2014 event. The
relative role of (i) deterministic evolution from initial
conditions versus (ii) stochastic forcing is evaluated by
integrating the model forward in time with either (i) the
deterministic component only [first term on the righthand side of (3); noise set to zero] or (ii) the stochastic
noise component only [second term on the right-hand
side of (3); initial state set to zero]. Comparing the observed SST anomalies with (i) the SST predicted by the
deterministic component or (ii) the noise-forced SST
evolution allows us to estimate the relative importance
of the noise forcing and deterministic dynamics for any
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given event. Note that since the noise forcing in this study
is calculated as a residual, by definition, it contains influences from both the stochastic noise forcing as well as
nonlinearities not captured by the linear approximation
to nonlinear dynamics. The importance of the timing for
the initial conditions and noise forcing is inferred through
initializing the linear inverse model (using the two terms
above) from different months, determined as the pentad
containing the first day of the respective month.
LIM also allows us to determine how CP and EP noise
forcing impacted each event. Similar to the method
described above, we integrate the linear inverse model
forward in time using staggered initial conditions.
However, by systematically removing the noise projecting onto either the CP or EP optimal, and comparing
the resulting SST forecast to the observed SST anomalies, we can determine the importance of CP or EP noise
forcing in generating the observed ENSO characteristics. We then quantify how well the LIM without either
the CP or EP noise forcing reproduces the observed SST
by calculating the root-mean-square error (RMSE) of
the CP index and EP index as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
un
u å (XLIM 2 XObs )2
t i51
,
(14)
RMSE 5
n
where XLIM is the CP or EP index in (1) for the LIM SST
forecast without CP or EP noise forcing, respectively;
XObs is the CP or EP index of the observed SST; and
n is the number of pentads. The RMSE is calculated
between the initialization (the first pentad of each month)
through the last pentad of the February following the
peak of the event. We use both CP and EP indices rather
than a single ENSO index (such as the Niño-3.4 index) in
order to capture the importance of the CP and EP noise
forcing in generating strong EP and CP spatial characteristics. It is important to note, however, that any given
ENSO event may be influenced by stochastic forcing that
is characterized by a combination, or complete lack of,
CP and EP forcing.
a. 1982/83 El Niño event
We first analyze the tropical Pacific SST anomalies
from the 1982/83 ENSO event. The Hovmöller diagrams
in Fig. 5 show the observed equatorial SST anomalies of
the event (top left) and various LIM simulations of the
SST anomalies. The remainder of the top row shows
the full LIM simulation [calculated using both terms on
the right-hand side of (3)], which match the observations
well no matter when the model is initialized, as expected
[this also serves as a check on the centered differencing
in (11)]. The middle row shows the SST anomalies due to
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the deterministic, or predictable, component of the LIM
[first term on the right-hand side of (3)] while the bottom
row shows the unpredictable, or noise forced, component, of the SST anomalies [second term on the righthand side of (3)], as explained above. By definition, the
dynamical SST anomalies plus the noise-forced SST
anomalies equal the SST anomalies produced by the full
LIM (i.e., top row 5 middle row 1 bottom row). The
label at the very top of each column indicates the month
of initialization; the model is initialized with observations from the first pentad of the month. The timing of
the initialization is indicated in the Hovmöller diagrams
with the horizontal black lines, and time increases along
the y axis and longitude (8E) are indicated along the
x axis.
Figure 5 indicates that deterministic dynamics do a
poor job at reproducing the observed SST anomalies for
the 1982/83 ENSO event prior to the October initial
conditions; instead, noise forcing dominates. Only when
the model is initialized in October or later does the dynamical forecast reproduce the peak amplitude of the
1982/83 ENSO event. This suggests the state of the
system in October, once the ENSO event has begun, is
sufficient for generating a skillful dynamical forecast of
the ENSO event, while the state of the tropical Pacific
prior to October was insufficient to generate a strong
ENSO event without the influence of stochastic forcing.
More specifically, these results suggest the noise forcing
that occurs after the August initialization but prior to
the October initialization is critical for producing the
October conditions that generate the large-amplitude
SST anomalies at the peak of the 1982/83 ENSO event.
Furthermore, once the deterministic dynamics begin to
dominate the ENSO development in boreal fall, they also
dominate the development of the La Niña event that
occurs during the following boreal winter. This shows that
the 1983/84 La Niña was a result of deterministic evolution from El Niño conditions in late 1982.
To determine whether the CP or EP noise forcing is
more important for the generation of the 1982/83 event, we
repeat the LIM simulations, except that we systematically
remove the CP or EP noise forcing from the model and
calculate the RMSE of the CP and EP index of the
resulting SST ‘‘forecast.’’ These results are summarized in
the RMSE calculations for the 1982/83 event shown in
Fig. 6a. The red plus (1) signs show the CP index RMSE
of the LIM run without CP noise forcing, while the black
crisscrosses (3) represent the EP index RMSE of the
LIM run without EP noise forcing. The large EP index
errors suggest EP noise forcing is critical to the development of the strong EP characteristics observed during
the 1982/83 event while CP noise forcing is much less
influential (indicated by small RMSE values for the CP
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FIG. 5. Hovmöller diagrams of equatorial Pacific (1208–2858E) SST (averaged from 28S to 28N) for the 1982/83 ENSO event. Time
increases upward along the y axis and longitude (8E) is indicated along the x axis. The observed SST anomalies are shown at the top left.
The remainder of the top row shows the SST anomalies predicted from the full LIM containing both deterministic dynamics and stochastic
forcing [calculated using both terms on the right-hand side of (3)]. (middle) The portion of SST anomalies predicted by the deterministic
dynamical component [first term on the right-hand side of (3)]. (bottom) The component of the SST anomalies driven by the stochastic
forcing term (second term on right-hand side of (3)]. The columns represent initializations every 2 months between February 1982 and
February 1983. The LIM is initialized using observations during the first pentad of the indicated month. The month of initialization is listed
at the very top of each column and is indicated by the horizontal black lines in the diagrams.
index). These results also show the noise forcing is
particularly important during boreal spring and summer. However, the magnitude of the EP errors decreases dramatically when the LIM is initialized in late
summer/fall, which supports the previous conclusions
that the state of the system during fall is sufficient to
develop into an ENSO event through deterministic dynamics alone, while the noise forcing during boreal spring
and summer is necessary to generate those conditions.
Finally, we analyze the structures of noise forcing during
the key months identified above for the development of
the 1982/83 EP El Niño. However, at any given time, a field
will contain contributions from the nondeterministic noise,
and from deterministic evolution (the coupled evolution of
ENSO). As such, the deterministic component needs to be
removed. Noise structures in fields that are not included in
the state vector [ynoise (x, y, t)] are identified using multiple
linear regression of the pentad NCEP–NCAR reanalysis
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FIG. 6. RMSEs calculated for CP (red 1 signs) and EP (black 3 symbols) indices for the LIM forecasts containing no CP or EP noise
forcing, respectively. The RMSE is calculated between the initialization month (indicated along the x axis) through the end of the
February following the peak of the event for the (a) 1982/83, (b) 1997/98, (c) 2009/10, and (d) 2015 El Niño events. (e) The RMSE values
for the failed 2014 event.
data (from 1982–2016) onto the noise time series [j(t) as
calculated in (11)]:
N
ynoise (x, y, t) 5
å ai (x, y)ji (t) ,
(15)
i51
where ai (x, y) are the multiple regression coefficients.
For each spatial point in a given field, we use multiple
linear regression to predict the full (1982–2016) time
evolution of the pentad data. The resulting dataset contains variability that covaries with the nondeterministic
component of the system: the noise. We then analyze the
monthly mean noise patterns for the specific months of
interest during the development of each ENSO event.
Figure 7 shows the monthly mean, uncoupled SLP (contours), 850-mb wind (vectors), and OAFLUX (shading) anomalies for July–October 1982. Since the October
1982 initial conditions are sufficient for generating
ENSO growth through deterministic dynamics alone, we
primarily focus on the stochastic forcing patterns in
the months prior to October. The July, August, and
September noise patterns show strong positive zonal
wind anomalies located in the western equatorial Pacific.
These zonal wind anomalies extend across much of the
equatorial Pacific during September 1982. These westerly
wind structures also match the JJA and SON EP noise
forcing patterns in Fig. 3.
b. 1997/98 El Niño event
We next analyze the atmospheric noise patterns that
occur during the development of the 1997/98 EP El
Niño. Figure 8 shows the same LIM simulations as Fig. 4,
but for the 1997/98 El Niño. Unlike the 1982/83 EP
event, which appears to have been largely influenced by
noise forcing in late boreal summer, the Hovmöller diagrams in Fig. 8 show that the deterministic dynamics of the LIM reproduce the observed evolution of
the 1997/98 El Niño more accurately from much earlier initializations. This agrees with previous studies
that also show a relatively long predictability of the
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FIG. 7. Monthly mean noise structures for July–October 1982. SLP anomalies (hPa; contours), 850-mb wind
anomalies (m s21; vectors), and OAFLUX anomalies (W m22; shading) are shown. Positive (negative) SLP
anomalies are indicated with red (blue) contours where the contour interval is every 1 hPa. The zero contour has
been omitted. OAFLUX fluxes are defined as positive upward.
1997/98 event (Newman and Sardeshmukh 2017). Our
results show this is especially true when the LIM is
initialized in June or later. Indeed, beyond June the
noise forcing has little to no effect on the amplitude of
the equatorial SST anomalies during the ensuing El
Niño event, though they do contribute to a more rapid
decay of the event and transition toward the 1998/99
La Niña.
Next, we determine if the noise forcing of the 1997/98
event is dominated by CP or EP noise. These results are
shown in Fig. 6b, which shows the RMSE of the CP and
EP index from the LIM forecasts without CP or EP
noise forcing, respectively. Again, the RMSE is calculated from the model initialization date through the end
of the February following the peak of the warming. The
LIM without CP noise forcing reproduces the observed
event well, as indicated by the small magnitudes of the
CP index RMSE (Fig. 6b; red plus signs), indicating CP
noise forcing is not important in the development of the
1997/98 event. The EP noise forcing, on the other hand,
appears to be much more important to the development
of the event, especially early in the year. The large
RMSE of the EP index for the LIM forecast without EP
noise forcing (Fig. 6b; black 3s) through May 1997
shows the importance of EP noise forcing early in the
year; however, the magnitude of the errors decreases
drastically beginning in early boreal summer. The sharp
decrease in RMSE that occurs after May 1997 indicates
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FIG. 8. As in Fig. 5, but for the 1997/98 EP El Niño event.
the EP noise during boreal spring is important for generating a state where deterministic dynamics dominate the
event development.
Figure 9 shows the monthly mean uncoupled SLP,
850-mb wind, and OAFLUX noise patterns associated
with the development of the 1997/98 event, focusing on
March–June 1997 in accord with the RMSE results
above. The noise forcing shows strong westerly wind
anomalies located along the equator in the central Pacific, during March and April of 1997, which match the
boreal spring (MAM) EP noise forcing structures in
Fig. 3. These results indicate that EP noise forcing,
particularly the positive zonal wind anomalies located
in the equatorial Pacific early in the year, was important
for the observed magnitude and strong eastern Pacific
characteristics of the 1997/98 ENSO event. In contrast to
the 1982/83 EP El Niño, this noise forcing occurred
much earlier, and hence the 1997/98 EP El Niño was
predictable with a longer lead time.
c. 2009/10 El Niño event
Figure 10 shows the role of deterministic dynamics
versus noise forcing though Hovmöller diagrams for the
2009/10 CP ENSO event, as before. For any lead time
the deterministic dynamics do a poor job at reproducing
the observed warming while the noise forcing dominates the development of the event. This shows the
importance of noise forcing to the development of the
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FIG. 9. As in Fig. 7, but for March–June 1997.
2009/10 CP event and its low predictability skill. Additionally, these results show, unlike the deterministic development of the La Niña events that occur after the
strong EP events described above, the La Niña event that
peaks during the boreal winter of 2010/11 is primarily
driven by noise forcing, even after the peak of the 2009/10
El Niño.
To identify the relative importance of EP and CP
noise forcing to the 2009/10 CP El Niño and determine
during which months the noise forcing is most important, we calculate the CP and EP index RMSEs of the
LIM forecasts run without CP and EP noise forcing,
respectively (Fig. 6c). The low RMSE values of the EP
index (Fig. 6c; black 3s) indicate the LIM run without
EP forcing has very little influence on the resulting SST
patterns, indicating the 2010 CP event was not highly
influenced by EP noise forcing. However, the LIM is
unable to accurately reproduce the observed event when
the CP noise forcing is removed from the model, especially when the model is initialized prior to June 2009.
The monthly mean structures of the uncoupled SLP,
850-mb wind, and OAFLUX noise that occurs between
March and June of 2009 are shown in Fig. 11. The SLP
patterns show slight NPO-like SLP anomalies located in
the north-central Pacific, which match the CP noise
forcing structures seen in Fig. 3. The dipole SLP structure
associated with the NPO is most apparent during April and
May; however, the negative SLP anomaly located in the
central North Pacific around 408N is evident in all four
months. The 850-mb wind structures show extratropical
wind anomalies associated with the SLP patterns, as expected. March and April also show strong positive zonal
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FIG. 10. As in Fig. 5, but for the 2009/10 CP El Niño event.
wind anomalies located along the equator in the central
Pacific. These zonal wind patterns closely match the CP
noise forcing structures shown in Fig. 3. Finally, the
OAFLUX patterns include downward heat flux anomalies in the subtropics, which project strongly onto the
PMM spatial structure. These OAFLUX patterns also
agree with the CP noise forcing structures in Fig. 3.
d. 2015/16 El Niño event
Figure 12 shows the same analysis applied to the
strong 2015/16 El Niño. Results show that deterministic
dynamics simulated the evolution of the 2015/16 event
well with very long lead times. More specifically, these
results show that the deterministic dynamics predict a
large El Niño event in early boreal spring, and capture
the full amplitude of the ensuing event in late boreal
summer (August). The noise forcing appears to be especially important during February 2015, and again in
July and August 2015, in generating the large-magnitude
ENSO event that was observed.
Figure 6d shows the RMSE results to quantify the
importance of CP noise forcing versus EP noise forcing to
the 2015/16 CP event. These results show small RMSEs
for both the CP and EP indices, except during February
and March of 2015, which contain relatively large EP
errors. The monthly composite SLP, 850-mb wind, and
OAFLUX noise forcing structures for February–May
2015 are shown in Fig. 13. The most striking noise forcing features are the large zonal wind anomalies that occur in the western and central equatorial Pacific during
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FIG. 11. As in Fig. 7, but for March–June 2009.
February and March. These zonal wind anomalies
strongly resemble the MAM EP noise forcing patterns
identified in Fig. 3 (right column). The Southern Hemisphere SLP, zonal wind, and OAFLUX anomalies in
March all strongly resemble the South Pacific Oscillation
of You and Furtado (2017), which is also evident in Figs. 3
and 4 (right columns). The monthly composites (Fig. 13)
also show noise patterns that project onto the previously
identified CP noise forcing patterns in Figs. 3 and 4. For
example, the OAFLUX during February contains a flux
into the ocean that resembles the PMM pattern seen in
Fig. 4 (left column).
These results show the 2015 ENSO event was largely
predictable in nature, due to the dominant role the deterministic dynamics played in the development. Furthermore, this event highlights the diversity of the noise forcing
for any given event. The 2015 event is difficult to classify as
either a CP event or an EP event as there is a significant
projection onto both CP and EP patterns (refer to the CP
and EP index time series in Figs. 2b and 2c, respectively).
This blend of CP and EP characteristics is apparent in the
influence of both CP and EP noise forcing structures.
e. 2014 nonevent
Finally, we analyze 2014 to see if the large-magnitude
event that many expected to occur at the end of 2014 was
‘‘ruined’’ by the role of the noise forcing. Most noteworthy, our results suggest no large-magnitude event
should have been expected during 2014. The deterministic dynamics, or the predicable component of the LIM,
do not forecast an ENSO event occurring at the end of
the 2014 calendar year, no matter when the model is
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FIG. 12. As in Fig. 5, but for the 2015 El Niño event.
initialized (Fig. 14). This suggests the state of the tropical
Pacific during 2014 never contained initial conditions
ideal for producing an ENSO event. Recall that forecasts
in Fig. 14 are initialized 12 months before those in Fig. 12,
so results are not directly comparable. Although the
monthly noise composites (Fig. 15) do show easterly wind
burst activity during June and July of 2014, which stalled
the development of a 2014 El Niño event according to Hu
and Fedorov (2016, 2018), the lack of an ENSO event
generated by the deterministic forecast using June initial
conditions suggests the WWB noise forcing in February
and March was insufficient to generate a state that could
develop into an event at the end of the year. Furthermore, our results support the conclusions of Chiodi and
Harrison (2017) that also suggest insufficient WWB activity to produce an ENSO event.
5. Conclusions and discussion
This study investigates the role of initial conditions
and noise in producing ENSO events with eastern
Pacific (EP) or central Pacific (CP) characteristics.
A forecast perspective is applied using linear inverse
modeling to separate the deterministic evolution of the
tropical ocean–atmosphere system from the noiseforced nondeterministic evolution. Within this forecast
framework, the noise plays two roles: it can enhance
deterministic growth by generating initial conditions
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FIG. 13. As in Fig. 7, but for February–May 2015.
with large projection onto optimal initial states or it can
lead to nondeterministic growth (forecast error) by
pushing the system away from the deterministic trajectory. While many studies have used LIM to identify
optimal initial conditions for EP and CP events, the
noise forcing capable of exciting those optimal structures has not been previously identified. This study
uses a LIM framework to identify specific noise structures capable of exciting CP and EP optimal initial
conditions and then analyzes the role of noise forcing
during the 1982/83 and 1997/98 EP ENSO events, the
2009/10 CP ENSO event, the 2015/16 event, and the
conditions during 2014.
We first use LIM to calculate the linear dynamics of
the tropical Pacific and identify the optimal initial conditions that maximize growth toward EP and CP events.
We then apply a centered-differencing method (Penland
and Hartten 2014) to calculate the noise forcing of the
tropical Pacific. By projecting the noise forcing onto the
CP and EP optimal initial conditions, we obtain time
series of how the noise pushes the system toward CP or
EP optimal initial conditions. These time series are then
used to investigate other fields to better understand the
structures within the noise forcing related to generating CP and EP growth. Prior studies show the importance of extratropical atmospheric forcing and the
seasonal footprinting mechanism in the development of
CP ENSO events (Yu and Kim 2011); the PMM structure of the CP optimal initial conditions found in this
study as well as the corresponding NPO structure of the
boreal winter stochastic forcing provide direct empirical
evidence in support of these findings. Further, the EP
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FIG. 14. As in Fig. 5, but for the failed 2014 event.
noise forcing identifies equatorial westerly wind activity
near 1508E and the South Pacific Oscillation (You and
Furtado 2017) as contributors to optimal EP initial
conditions.
The LIM framework is then used to investigate the
relative roles of initial conditions versus stochastic
forcing in the evolution of specific ENSO events. The
LIM is integrated forward from specific times along (i) a
deterministic trajectory in which the initial conditions
are integrated forward with zero noise forcing and (ii) a
nondeterministic trajectory in which the initial conditions are set to zero and the LIM is integrated forward
using only the estimated noise forcing. The technique
accurately captures diversity in the timing and spatial
structure of noise forcing although the noise forcing is
convolved with the estimated deterministic dynamics.
Diversity in the timing of the initial conditions and
noise forcing is especially evident when comparing the
1982/83 and 1997/98 EP El Niño events. Results show
that positive zonal wind anomalies play a significant role
during the development of both the 1982/83 and 1997/98
EP ENSO events, although the two events evolve under
very different conditions. Similar to the conclusions of
Takahashi and DeWitte (2016), the large amplitude of
the 1982/83 event was related to the strong equatorial
wind anomalies in late boreal summer (August and
September). We also show that the state of the tropical
Pacific prior to October 1982 was not conducive to
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FIG. 15. As in Fig. 7, but for February, March, June, and July 2014. Only the months with large-magnitude noise
forcing structures are shown.
producing a large event via the deterministic trajectory.
It is possible the limited observations may play a role in
the low predictability of the 1982/83 event.
In contrast, the deterministic trajectory for the 1997/
98 El Niño was predictable using observed initial conditions as early as late winter 1997 (February). Large
equatorial westerly wind anomalies in March and April
1997 established a tropical Pacific state that was sufficient for producing a large-amplitude El Niño event by
late 1997. Hong et al. (2014) find extreme-magnitude El
Niño events (i.e., 1982/83 and 1997/98) are preceded by
enhanced low-level westerly wind anomalies in the westcentral Pacific. Furthermore, Harrison (1984) shows a
southerly jet prior to the occurrence of the sustained
westerly wind anomalies in 1982 may also play a role in
the development of the large-magnitude event observed.
Beyond May 1997, noise forcing played very small role in
the evolution of the 1997/98 El Niño event. Furthermore,
the La Niña conditions following the two extreme El Niño
events of 1982/83 and 1997/98 are shown to largely result
from the dynamical evolution of the system from their
preceding El Niño events.
Analysis of the 2009/10 CP ENSO event highlights
the role of the stochastic forcing in producing some
ENSO events. The 2009/10 event was largely noise
driven throughout the entirety of the event. Only
during the late boreal fall of 2009 did the deterministic
trajectory capture the amplitude of the event. CP noise
forcing during April and May 2009 was especially important for the development of CP characteristics for
the resulting event. Further, unlike the 1983/84 and
1998/99 La Niña conditions, the 2010/11 La Niña event
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was largely unpredictable while the 2009/10 El Niño
event itself was still developing. However, these results show that the 2010 La Niña event contained
some deterministic predictability once the El Niño
event reached its peak magnitude, as suggested by Kim
et al. (2011).
Not all events evolve along CP or EP trajectories, as
evidenced by the 2015/16 event. Strong EP forcing (especially equatorial westerly winds) in February and
March 2015 was especially important for setting the
stage for the ensuing event. Analysis shows that the
highly anticipated 2014 event, which never materialized,
was relatively well predicted by the deterministic trajectory throughout 2014.
Our method of estimating noise forcing allows the
statistics of the system to determine the assumed dynamics (the LIM), so that we can directly estimate the
noise as a residual. An alternative approach for determining noise forcing of ENSO would be to first assume a theoretical model for ENSO variability, and
then calculate the additive and multiplicative components of the noise forcing using a predetermined relationship between the system state and the noise,
such as by using linear regression to calculate coefficients (Levine and Jin 2017; Levine et al. 2016). Our
method includes both the additive component and
possibly some contribution from the multiplicative
component of the noise [see Martínez-Villalobos et al.
(2018) and Sardeshmukh and Sura (2009) for further
description of the multiplicative noise contribution to
the dynamical operator and noise amplitude], but it
does not distinguish between these components. We
note that Levine et al. (2016) find modest contributions of the multiplicative noise component only during
extreme ENSO events, which should be taken into
consideration in interpreting the results in section 4.
The technique used in this study has some other limitations. For example, the linear approximation in LIM
assumes that the system’s nonlinear dynamics have
much shorter memory (are more rapidly decorrelating)
than the linear dynamics. Where and when this assumption is not valid, the LIM will not capture all the
nonlinear dynamics properly. [Of course, many CMIP5
models also may not correctly represent nonlinear dynamics, as shown for example in Karamperidou et al.
(2017).] LIM is known to have reduced skill in the far
eastern Pacific (Newman and Sardeshmukh 2017),
which could well be due to unresolved nonlinearity, as
well as to strong seasonality that is not included in the
linear operator. The relatively short record used to train
LIM could allow some deterministic and noise effects to
be convolved, especially given the reliance on a truncated EOF space to limit the dimensionality of the
predictable dynamics. Still, tests on the validity of LIM
in the tropical Pacific herein and in previous studies
show LIM performs well in the tropical Pacific (e.g.,
Penland and Sardeshmukh 1995). Also, Newman and
Sardeshmukh (2017) suggested that, since both LIM and
multimodel ensemble mean forecasts generally outperform individual CGCMs in the equatorial Pacific,
essentially unpredictable nonlinear interactions cause
random forecast errors that are averaged out better
in multimodel ensemble mean forecasts than in the
individual-model ensemble mean forecasts, in agreement with the assumption underlying LIM. Moreover,
Chen et al. (2016) show a nonlinear inverse model
(NLIM) does not improve tropical Pacific SST forecast results over the regular LIM, and Ding et al.
(2018) suggest that most of the central tropical Pacific
skill within some NMME models is effectively linear.
Further research is being done to test our findings in
a coupled general circulation model using both the
methodology herein, as well as directly applying the
noise forcing structures as an external forcing to a
coupled general circulation model.
While the present study highlights some interesting
cases, it is important to note the development of any
given ENSO event can be influenced by a rich variety of
initial conditions and noise structures. Further, these
noise structures themselves are a blend of a variety of
phenomena that can excite growth in the tropical Pacific.
While the role of noise forcing for ENSO development
varies greatly for each event and characterizing the structures within the noise forcing is challenging, the present
study provides a single framework for parsing through the
myriad processes that contribute to ENSO diversity.
Acknowledgments. We wish to thank the reviewers for
providing helpful comments. This work was supported
by NSF Climate and Large Scale Dynamics Project
1463970.
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