1. Introduction
El Niño–Southern Oscillation (ENSO) is the dominant mode of tropical variability on interannual time scales. It consists of an irregular oscillation of sea surface temperature anomalies over the tropical eastern Pacific Ocean, with a periodicity in the 3–7-yr range. Through atmospheric teleconnections, ENSO impacts weather across the globe and is the leading source of skill for seasonal and interannual forecasts. It is imperative that general climate models (GCMs) capture this source of predictability.
Despite the importance of ENSO, state-of-the-art coupled climate models show a wide range of ENSO behavior (Guilyardi et al. 2009; Flato et al. 2013), with differences not only from model to model, but even from one model version to the next. In general, the models in phase 5 of the Coupled Model Intercomparison Project (CMIP5) show large deficiencies in ENSO amplitudes, spatial structures, and temporal variability (Flato et al. 2013).
Many authors have demonstrated the importance of small-scale atmospheric variability in ENSO dynamics (Penland and Sardeshmukh 1995; Flügel et al. 2004; Yeh and Kirtman 2006). The low resolution of current coupled climate models (of order 1°–2°) is inadequate for capturing this small-scale variability. Increasing the resolution has been shown to improve the representation of ENSO (e.g., Small et al. 2014) but this remains often too computationally expensive for multidecadal climate projections.
An alternative to represent unresolved subgridscale variability is through the use of stochastic parameterization schemes (Palmer 2001). Models with such schemes represent subgridscale atmospheric processes as a predictable deterministic plus an unpredictable stochastic component. Stochastic parameterizations have been widely used in the weather and seasonal forecasting community because of their beneficial impact on probabilistic forecast reliability (e.g., Berner et al. 2009; Leutbecher et al. 2017; Weisheimer et al. 2014). Despite their beneficial impact on weather and seasonal predictions, the use of stochastic parameterizations in climate models remains a scientific frontier (Berner et al. 2017), since model error on longer time scales tends to be dominated by deterministic rather than random model error.
Of particular relevance to this study are the findings by Christensen et al. (2017), who demonstrated that stochastic perturbations to the atmospheric component of the Community Climate System Model, version 4 (CCSM4), leads to remarkably improved ENSO variability in the model. Without stochastic perturbations, the model ENSO is too regular and too strong. Including the stochastic perturbations improves the power spectrum of SSTs in the Niño-3.4 region [Fig. 1, after Christensen et al. (2017)] and the variance of monthly SST and wind anomalies is also in better agreement with observations [Fig. 2, cf. Figs. 5 and 6 of Christensen et al. (2017)]. Note that Fig. 2 shows the variance difference of a control simulation (CNTL) from ECMWF Re-Analysis of the 20th-Century Climate (ERA20C) indicating that the variance in the control simulation is too strong. The stochastically perturbed parameterization tendency (SPPT) reduces the variance leading to a better agreement with the reanalysis.
Power spectrum of the Niño-3.4 index, defined as the monthly SST anomaly averaged over 5°S–5°N and 170°E–120°W, for HadISST2 observations (black), CNTL (blue), and SPPT (red). The top axis indicates period in years, and the bottom axis indicates frequency in cycles per month. The shading denotes the spectral range obtained by sampling realizations from LIMs fitted to CNTL, SPPT, and HadISST (see text). After Christensen et al. (2017).
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
Variance of monthly anomalies over the period 1900–2010 for (a) ω500, (d) u850, and (g) SST in ERA20C. (b),(e),(h) Difference in variance between ERA20C and CNTL, where CNTL spans the years 1870–2004. (c),(f),(i) Difference in variance between SPPT and CNTL for the period 1870–2004. Note that the contour interval in (c), (f), and (i) is half that of (b), (e), and (h).
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
Christensen et al. (2017) proposed a number of possible mechanisms for this improvement, but a complete understanding was outside the scope of that study. In the present study, we will reexamine these coupled simulations. Our aim is to understand how fast fluctuations on weather time scales (as mimicked by the stochastic parameterization) can impact interannual tropical variability. In particular, we will address the mechanisms by which perturbations to the atmosphere can
- reduce SST and wind variability, and
- broaden the spectrum of tropical SSTs, that is, increase ENSO irregularity.
To gain insights into the dynamical mechanisms leading to the change in ENSO variability, we fit linear inverse models (LIM) to the two climate simulations and subsequently compare the LIMs governing ENSO dynamics and its weather forcing. Linear inverse models (Xu and von Storch 1990; Penland and Magorian 1993; Latif et al. 1994; Wu et al. 1994; Balmaseda et al. 1995) have been demonstrated to show excellent skill in predicting tropical SST variability (Penland and Sardeshmukh 1995; Newman et al. 2009, 2011; Alexander et al. 2008; Newman and Sardeshmukh 2017) and have also been applied to observed Atlantic sea surface temperatures (Zanna 2012). In particular, the least damped oscillatory eigenmode, or principal oscillation pattern (POP; von Storch et al. 1988, 1995; Penland 1989) captures the basic features of ENSO (Penland and Magorian 1993; Penland and Sardeshmukh 1995; Gehne et al. 2014).
Kleeman (2011) performs a comprehensive spectral analysis for a generalized class of stochastic models including LIM and applies it to the two-dimensional stochastically forced oscillator, which can be interpreted as ENSO POP. Different from previous studies, which mainly focus on using LIMs as predictive models, we exploit their utility here also as a general tool to investigate model differences and model sensitivities (Shin et al. 2010).
Given the underlying assumptions, LIMs are fundamentally limited to which degree they can model non-Gaussian and nonlinear features. For example, LIMs cannot capture the apparent greater persistence of La Niña versus El Niño events (DiNezio and Deser 2014). In their simplest form, the noise is assumed to be state independent, the resulting distributions Gaussian, and thus unable to capture the skewness in, for example, the distribution of the Niño-3.4 index (Burgers and Stephenson 1999). However, the inclusion of state-dependent noise terms enables LIMs in principle to model skewed distributions (Sardeshmukh and Sura 2009).
The paper is organized as follows: the linear inverse modeling approach is summarized in section 2. A simple damped linear oscillator forced by white noise is used to illustrate the impact of perturbing the damping rate and frequency of the eigenmodes of a linear system. Datasets and the setup for the numerical experiments are introduced in section 3. Section 4 contains the results of fitting LIMs to the coupled climate simulations, followed by a discussion (section 5) and the conclusions (section 6).
2. Methodology and simple example
a. Damped linear system forced by additive white noise
b. Damped linear system forced by additive and multiplicative white noise
c. Example: The perturbed 2D-damped harmonic oscillator forced by additive white noise
(a) Sample time series of damped linear oscillators driven by additive and perturbed by multiplicative white noises. Oscillator with parameters damping rate 
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
The derivation reveals that the multiplicative noise introduces two terms in the denominator: the noise-induced drift and an additional term to the noise covariance matrix. For perturbations to the frequency, these two terms are of equal magnitude but opposite sign, so that they cancel each other. For perturbations to the damping rate, they have the same sign, leading to the 
Physically, the increase in variance—or energy—in the case of perturbing the damping rate can be understood by visualizing a pendulum. At maximum amplitude, perturbations increasing the amplitude will increase the potential energy, while perturbations toward the state of rest will increase the kinetic energy, both leading to an increase in energy and thus variance.
While the stabilizing influence of noise has been discussed in the geophysical contexts (e.g., by Sardeshmukh et al. 2001, 2003), this is to our best knowledge the first explicit application to an oscillator.
d. Spectra
3. Model description and experiments
The numerical simulations of the coupled ocean–atmosphere system analyzed in this work are conducted with the CCSM4 developed by the National Center for Atmospheric Research (NCAR) and cover the period 1870–2004 (135 years). The atmospheric component is the Community Atmosphere Model, version 4 (Gent et al. 2011). All simulations were conducted with the finite-volume dynamical core at a resolution of 0.9° × 1.25° with 26 vertical levels and the use of observed carbon dioxide concentrations in the atmosphere. The ocean is simulated by the Parallel Ocean Program, version 2 (Danabasoglu et al. 2012), with 1.0° horizontal resolution and 40 levels in the vertical. It is also actively coupled to the Community Ice Code, version 4 (Hunke and Lipscomb 2008), at 1.0° resolution and the Community Land Model, version 4 (Lawrence et al. 2011).
This study focuses on three variables: sea surface temperature (SST), zonal wind at 850 hPa (u850), and vertical pressure velocity (or p velocity) at 500 hPa (ω500) in the tropical belt between 20°S and 20°N. The vertical p velocity 
For comparisons with the observed record, we use ERA20C, which spans the years 1900–2010. ERA20C assimilates surface pressures and surface winds over the oceans and uses sea surface temperatures from HadISST2 as lower boundary conditions (Poli et al. 2016). Because of the limited number of observations used, there are better reanalysis products available for the atmospheric state, especially for the better observed period since 1979. However, the consistency between the atmosphere and SSTs over an extended period makes the use of ERA20C attractive. We keep in mind that—especially when it comes to the vertical p velocity—the reanalysis will be dominated by the model first guess rather than observations.
For data-reduction purposes all variables of each dataset were standardized with the spatially averaged standard deviation of ERA20C and projected onto the empirical orthogonal functions (EOFs) computed from ERA20C. The EOFs were computed for each variable separately and the anomalies subsequently projected onto the first 10 EOFs of ω500 and u850 and the first 30 EOFs for SST. The LIMs are fitted to the state consisting of the combined principal components of ω500 and u850 and SST. These variables were chosen to capture the interplay between atmospheric and oceanic key players relevant to ENSO rather than construct the most skillful LIM. The LIM is primarily used to study the impact of adding stochastic tendency perturbations to the atmospheric component, while the ocean component was unchanged. Hence, we did not include a subsurface ocean variable, which might have added forecast skill (Newman et al. 2011). While there is some sensitivity to the details of the standardization as well as the number of EOFs retained, all findings reported here hold qualitatively over a wide range of choices.
4. Results
a. Covariance evolution in model and reanalysis
In this section we will demonstrate the LIMs’ ability to capture the basic features of tropical variability. We will then investigate the LIMs on a mode-by-mode basis to gain insight on the mechanisms, by which SPPT affects the ENSO mode. To evaluate the model simulations and LIM, we examine the covariance evolution of ω500, u850, and SST in the tropical belt.
For ERA20C, the largest SST variance is in the tropical east and central Pacific with a pattern reminiscent of El Niño (Fig. 4). The variance swings from a positive to negative anomaly with decreasing amplitude, suggestive of a damped oscillation with a period of 4 years or so. The SST signal is accompanied by an atmospheric oscillation with the same period. For u850, the pattern of maximal variance is shifted to the western tropical Pacific with a secondary maximum over the Indian Ocean. The covariance of ω500 is boomerang shaped with the two arms extending just north and south of the equator across the Pacific basin.
Diagonal of the autocovariance matrix at lags 
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
For the simulations CNTL and SPPT, the centers of maximal variance are in the same east and west Pacific locations (Figs. 5, 6) indicating that the characteristic features of ENSO variability are captured by the climate model. The amplitudes in SPPT are notably smaller for all time lags, leading to a better agreement with those of ERA20C. This indicates that the oscillation in SPPT is more damped than in CNTL.
Diagonal of lagged autocovariance matrix for SSTs in simulations CNTL and SPPT and lagged autocovariance matrix as predicted by a LIM fitted to CNTL and SPPT using 1-month lag covariances. Contour intervals are as in Fig. 4.
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
As in Fig. 5, but for u850.
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
Next, linear inverse models in (5) are fitted by estimating the covariance matrix of each experiment for a time lag of 
The LIM captures the characteristic features of the covariance evolution in the climate simulations remarkably well (Figs. 5, 6). For brevity, the covariances are shown here only for SST and u850. The evolution of 
As scalar metrics we compute the spatially averaged autocorrelations and autocovariances over the tropical band (Fig. 7). Consistent with the maps, the spatially averaged autocorrelation has the typical signature of a damped oscillation. We note that the autocorrelation curves for the SPPT simulation are overall in much better agreement with those in ERA20C. The LIM predicts the autocorrelation and autocovariance remarkably well up to lags of at least 
(a) Autocorrelation and (b) autocovariance of SSTs averaged over the tropical band between 20°S and 20°N. Solid lines are for ERA20C (black), CNTL (blue), and SPPT (red); dashed lines are for LIM predictions. The lag-0 variance is denoted by circles. Autocorrelations for (c) ω500 and (d) u850 are shown.
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
b. Linear inverse model eigenmodes
Next, we will investigate if the differences in the LIMs can be used to understand why SPPT has a better representation of ENSO variability. Since 
For each eigenmode of 
Eigenvalues of the LIM (a) linear feedback matrix 
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
The spatial pattern of this mode is shown for each component 
(left) Real and (right) imaginary parts of the least damped oscillatory eigenmode (or principal oscillation pattern) corresponding to the mature phase of ENSO in the left panels and the characteristic precursor pattern in the right panels. Components of the mode are shown for ω500, u850, and SST for (a)–(f) ERA20C, (g)–(l) CNTL, and (m)–(r) SPPT. Decorrelation time 
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
The SST component of the least damped oscillatory eigenmode has the highest amplitudes in the central and east Pacific, with a pattern highly reminiscent of El Niño (Fig. 9). The u850 component has a pronounced dipole pattern with a negative center over the Maritime Continent and Indian Ocean and a positive center over the western Pacific (Figs. 9c,i,o), indicative of anomalous low-level wind divergence over the Maritime Continent. This is consistent with the anomalous subsidence over the Maritime Continent and anomalous convection over the central Pacific (Figs. 9a,g,m), as typically observed during the warm phase of ENSO.
The imaginary part of the least damped oscillatory eigenmode shows anomalous warming in the east Pacific, but as reported in other studies (Penland and Sardeshmukh 1995; Gehne et al. 2014), the anomalies are much weaker and confined to a narrow tropical band (Figs. 9f,l,v). The corresponding wind pattern shows anomalous westerlies over the Maritime Continent and western Pacific (Fig. 9d) and is highly correlated with the second EOF of u850 (not shown). The imaginary part precedes the peak pattern by a quarter of a period, which amounts to 11 months for ERA20C, 12 months for CNTL, and 13 months for SPPT. The peak pattern is then followed by the negative precursor pattern after another 12 months or so and will then develop into the La Niña peak pattern after another quarter period.
The model captures the main characteristics of the peak ENSO pattern with anomalously warm SSTs over the east Pacific corresponding to subsidence and low-level wind divergence over the Maritime Continent. While the ENSO peak and precursor patterns in the model simulations agree remarkably well with those from ERA20C, there are pronounced differences in the associated eigenvalues (Fig. 8a). In particular, the decorrelation time in SPPT is with 
Given that the ENSO POP has the form of a damped oscillator, we return to the analytical results from section 2. From the perturbation experiments we know that a decrease in decorrelation time—as in experiment SPPT—is consistent with perturbations to the frequency of an oscillator. Indeed, the parameters ν and ω in Fig. 3 were chosen to produce an oscillation with a period of 
While the simple model explains the mechanisms, by which stochastic perturbations to the atmosphere can change the spectrum of ENSO, it does not provide an explanation for the observed reduction in variance (Fig. 2). Recall that perturbations to the frequency did not result in a change to the equilibrium variance in (14). We will return to this question after analyzing the second LIM component—the noise covariance matrix.
c. Eigenanalysis of noise covariance matrix 
Next, we have a closer look at the noise covariance matrix 
Maps of the noise variance given as diagonal of 
Diagonal elements of the noise covariance matrix for model experiments (a),(d),(g) ERA20C; (b),(e),(h) CNTL; and (c),(f),(i) SPPT. Maps for variables (top) ω500, (middle) u850, and (bottom) SST are shown.
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
d. Impact of differences in feedback and noise covariance matrices
Our results suggest that the stochastic perturbations of SPPT introduce a noise-induced drift resulting in a stabilization of the feedback matrix. Returning to the simple oscillator example in (5), we saw that multiplicative noise resulted in an additional, positive definite term in the effective noise covariance matrix 
(a),(d),(g) Diagonal elements of the difference between LIM covariances 
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
Including only the changes to the feedback matrix, 
Overall, the increase of equilibrium variance in experiment exp2 is weaker than that variance decrease of exp1, which suggests that the noise-induced stabilization in 
5. Discussion: Climate simulations forced by repeating climatological annual cycle of SST
a. Variability of monthly anomalies
We have hypothesized here that the perturbations introduced by SPPT to the atmospheric component of CCSM4 perturb the frequency of the El Niño–Southern Oscillation, resulting in a noise-induced drift that enhances the damping of ENSO. To further test this hypothesis, we ask what the effect of SPPT would be if there was no ENSO. The simple model suggests that if there is no oscillation, there is no frequency to perturb and presumably no stabilizing noise-induced drift. To confirm this, we performed two additional 10-yr uncoupled atmospheric GCM integrations with and without SPPT using the atmospheric component of CCSM4 with a prescribed repeating climatological annual cycle of SSTs as boundary forcing. By experimental design, there is no ENSO, indeed no interannual SST variability, in these uncoupled integrations.
Comparing the variance of zonal wind and vertical p velocities in the two simulations shows that their differences are very small (Fig. 12). In most places, the differences are not statistically significant at the 95% confidence level, although there is a region in the western Pacific just north of the equator, where SPPT might increase the monthly variability in these variables. This confirms that SPPT does not reduce the interannual wind variability in the absence of interannual SST variability.
Difference in variance of monthly anomalies of (a) ω500 and (b) u850 in 10-yr-long simulations forced by repeating climatological SSTs. Shown is the difference between simulations with and without SPPT. Note that the contour interval in (a) is one-fourth and in (b) it is half of that in the respective panels in Figs. 2c and 2f. Differences that are statistically significant at the 95% confidence level are stippled. (c) Histogram of 5-day running mean zonal wind anomalies in CNTL (blue) and SPPT (red). Westerly wind bursts are defined as zonal wind anomalies exceeding 5 m s−1 (black vertical line).
Citation: Journal of Climate 31, 20; 10.1175/JCLI-D-18-0243.1
This is consistent with the results of Christensen et al. (2017), who also performed uncoupled atmospheric simulations, but with observed interannually varying SSTs as lower boundary forcing. These simulations have the ENSO signal in the lower boundary forcing, but no freely evolving coupled ENSO mode. This study found that introducing SPPT reduced the atmospheric variability, but the change was less pronounced than in the fully coupled simulation. In conclusion, the model must have a freely evolving coupled ENSO mode for SPPT to have a strong impact on interannual variability.
b. Westerly wind bursts
Previous studies demonstrated that extreme westerly wind anomalies over the equatorial western-central Pacific—so-called westerly wind bursts (WWBs)—can play a key role in the outbreak of El Niño events, since they have the potential to trigger eastward-propagating oceanic Kelvin waves, which lead to a warming in the equatorial central and eastern Pacific (e.g., McPhaden and Taft 1988; Yu and Rienecker 1998; Lengaigne et al. 2004). The state-dependent nature of the stochastic WWB forcing is thought to be particularly important (Jin et al. 2007; Levine and Jin 2010, 2017). A change of either the magnitude of WWBs or their state dependence would constitute a physical mechanism by which SPPT can influence the development and evolution of ENSO. Following Levine and Jin (2017), Christensen et al. (2017) analyzed the state dependence of WWB and found indeed a reduction in the state dependence of WWB in the simulations with SPPT. However, in coupled simulations it is extremely difficult to disentangle the causality of WWBs and SSTs in the west Pacific, since on one hand wind anomalies are thought to be the primary forcing for ocean variability, but on the other are modulated themselves by the SSTs (Tziperman and Yu 2007).
To see if the magnitude of WWBs has been changed independently of the state dependence, we analyze the distribution of zonal wind anomalies in our simulations with climatological SSTs. We take wind anomalies as the deviation from the daily climatology (Harrison and Vecchi 1997) and subsequently compute the running 5-day mean to pick out longer lasting events. WWBs are defined as events, where the smoothed anomalous zonal winds exceeded 5 m s−1. Histograms of the zonal wind anomalies show that SPPT reduces extreme easterly as well as westerly wind anomalies (Fig. 12c). In particular, the frequency of WWBs is reduced.
Importantly, this change in the tails of 5-day running means does not imprint on the standard deviation of monthly mean anomalies (Fig. 12b). We conclude that in addition to a reduced state dependence (Christensen et al. 2017), the reduced occurrence of strong WWB in SPPT might play a key role in reducing ENSO variability. Future work will be targeted at understanding why SPPT reduces the extreme zonal wind anomalies.
6. Conclusions
With the aim of understanding changes in ENSO irregularity, we fitted a linear inverse model to climate simulations with and without the stochastic parameterization scheme SPPT. In particular, we set out to understand the dynamical mechanisms by which perturbations to the atmosphere reduced tropical SST and wind variability and decreased the decorrelation time of ENSO variability.
In the experiment with stochastic perturbations, the least damped oscillatory eigenmode or principal oscillation pattern (POP) is characterized by a more damped oscillation, reducing the decorrelation time of the mode from 17 to 11 months, which according to the Wiener– Khinchin theorem explains the broadening of the spectrum (Fig. 1). This is an improvement in comparison to the 20th-century reanalysis from ECMWF, in which this mode has a decorrelation time of about 8 months.
The fact that this least damped coupled mode has a pattern and period characteristic for ENSO supports the theory of ENSO as a damped oscillation of the coupled ocean–atmosphere system forced by stochastic atmospheric noise (Penland and Sardeshmukh 1995; Kleeman and Moore 1997; Moore and Kleeman 1999; Gehne et al. 2014). Note that independent of the nature of ENSO, each POP has by definition the form of a damped oscillator (von Storch et al. 1988; Penland 1989; Kleeman 2011), so that differences among LIMs can more generally be used to diagnose model differences and differences between model and nature.
Turning to the simple model of a stochastically damped linear oscillator, we studied the effect of perturbing its frequency versus its damping rate. We found that perturbations to the damping rate result in increased variance and a more peaked spectrum, whereas perturbations to the frequency reduce the temporal memory and broaden the spectrum (Fig. 3). The GCM results consisting of a decrease in decorrelation time and broadening of the spectrum are thus consistent with perturbations to the frequency of the least damped eigenmode of the LIM.
The perturbations to the frequency result in further stabilizing the LIM’s feedback matrix. This stabilization is due to the so-called noise-induced drift. The stabilizing influence of noise in geophysical contexts has been discussed (e.g., Sardeshmukh et al. 2001, 2003) for the example of Rossby waves in a stochastically fluctuating medium. In our ENSO context, we have shown that the noise-induced stabilization of the dynamical operator is sufficiently strong to reduce the variances of SST and winds. The noise-induced variance reduction has a pattern similar to the difference between SPPT and CNTL, but weaker magnitude.
Our results also demonstrate some of the complexities of coupled Earth-system modeling and tuning. The effect of the stochastic perturbations—used here as proxy for fast-physics processes—lie in the modulation of a slow process, here the El Niño–Southern Oscillation. The effect can therefore only be studied when fully interacting with this slow process (i.e., in a fully coupled modeling framework), and was not evident in the statistics of monthly anomalies in atmospheric simulations with repeated climatological SST forcing.
While a linear inverse model cannot replace coupled climate models, analyzing the changes to the operator of the LIM on a mode-by-mode basis allows insights into the dynamical mechanisms, by which stochastic perturbations to the atmospheric component with time scales of 6 h can impact tropical interannual climate variability.
Future work will focus on physical mechanisms by which fast-physics processes impact ENSO, starting with an in-depth analysis of the impact of SPPT on westerly wind bursts. Initial results pointed to a reduction of the occurrence and magnitude of westerly wind bursts as well as a reduced state dependence (Christensen et al. 2017), which will be investigated further.
We thank Dr. Maria Gehne, Prof. Natalie Burls, and an anonymous reviewer for their efforts in reviewing this manuscript. Thanks also to Dr. Grant Branstator whose insightful comments improved an earlier version of this manuscript. We acknowledge numerous delightful discussions with Dr. Cecile Penland. In particular, she was the first to mention to the first and third authors that perturbations to an oscillator can stabilize the system. Thanks also to Dr. Justin Small for sharing his results on the impact of increased horizontal resolution on tropical variability and Danielle Coleman for running the coupled climate simulations. The CESM project is supported by the National Science Foundation and the Office of Science of the U.S. Department of Energy. SPPT in CESM was developed in part under EPA Grant RD-83520501.
APPENDIX A
Fitting a Linear Inverse Model
The theory presented in this section makes the assumption that the noise is white (i.e., that it is δ correlated in time). However, the stochastic parameterization in the GCM is written as a red-noise process with a decorrelation time of 6 h. More generally, even fast physical processes tend to be continuous and hence are never δ correlated. Generally speaking, it is still possible to describe such a system stochastically, as long as the decorrelation time of the finitely correlated fast processes are much smaller than those governing the system dynamics (Horsthemke and Levefer 1984; Penland 2003). An example of the impact of generating stochastic perturbations with a red rather than white spectrum is given for the example of the Rossby wave response in Sardeshmukh et al. (2003).
APPENDIX B
Derivation of Covariance Matrix for a Linear Stochastic Process
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Note again that ν refers to the damping rate and 
