## Abstract

The seasonal variability of 300-hPa global streamfunction fields taken from a 40-yr period of reanalyzed observations starting on 1 January 1958 and from long 497- and 900-yr general circulation model (GCM) datasets forced by sea surface temperatures (SSTs) is examined and analyzed in terms of empirical orthogonal functions (EOFs), principal oscillation patterns (POPs), and particularly finite-time principal oscillation patterns (FTPOPs). The FTPOPs are the eigenvectors of the propagator, over a 1-yr period covering the annual cycle, that has been constructed by fitting a linear stochastic model with a time-dependent matrix operator to atmospheric fluctuations based on the daily or twice-daily 300-hPa streamfunction datasets.

The leading FTPOPs are large-scale teleconnection patterns and by construction they are the empirical analogs of finite-time normal modes (FTNMs) of linear instability theory. Hence, by comparing FTPOPs to FTNMs, the study provides insight into the ability of linear theory to explain seasonal and intraseasonal variability in the structure and growth rates of large-scale disturbances. The study finds that the leading FTPOP teleconnection patterns have similar seasonal cycles of relative growth rates and amplitudes to the leading FTNMs of the barotropic vorticity equation with 300-hPa basic states that change with the annual cycle; the largest amplitudes of both theoretical and empirical modes occur in late boreal winter or early spring, and minimum amplitudes in boreal autumn, with the GCM-based FTPOPs having additional secondary maxima in early boreal summer. In each month, there are leading POPs and EOFs that closely resemble the leading FTPOPs. Also, the growth rates of leading FTNMs and FTPOPs during each season are generally similar to those of respective leading normal modes and POPs calculated for that season. Thus the perturbations are reacting to the seasonally varying basic state faster than the state is changing and this appears to explain why linear planetary wave models with time-independent basic states can be useful. Nevertheless, *intermodal* interference effects, as well as *intramodal* interference effects, between the eastward and westward propagating components of single traveling modes, can play important roles in the evolution of FTPOPs and FTNMs, particularly in boreal spring.

This study has examined the roles of internal instability and interannual SST variability in the behavior of leading FTPOPs and has also used comparisons of FTPOPs and FTNMs for GCM simulations with and without interannually varying SSTs to assess the role of internal instability and SST variations in organizing interannual atmospheric variability. The comparison indicates that both factors are significant. The results found here also support a close relationship between the boreal spring predictability barrier of some models of climate prediction over the tropical Pacific Ocean and the amplitudes of large-scale instabilities and teleconnection patterns of the atmospheric circulation.

## 1. Introduction

During the last two decades, normal mode instability theory, generalized to basic states with horizontal and vertical variations, has been employed to examine the generation mechanisms and dynamical properties of the major classes of synoptic- and large-scale atmospheric disturbances. Explanations have been proposed for the causes of localized cyclogenesis and for the location of the major storm tracks and their deflection during blocking; and modal structures have been proposed as theoretical counterparts for blocking and large-scale teleconnection patterns, intraseasonal oscillations, and tropical disturbances. In particular, the roles of baroclinic and barotropic instability in the formation of blocking and other low-frequency anomalies were examined by Frederiksen (1982, 1983) and Simmons et al. (1983). Modal structures associated with onset-stage baroclinic and mature-stage barotropic processes in the formation of Pacific–North American anomalies were found to compare closely with the observations of Dole (1983, 1986), Dole and Black (1990), and Evans and Black (2003). Other large-scale modes were found to be remarkably similar to east Atlantic and North Atlantic Oscillation teleconnection patterns (Wallace and Gutzler 1981; van Loon and Rogers 1978). The roles of normal mode growth in the formation of large-scale low-frequency anomalies were further analyzed in the works of Branstator (1985, 1987), Legras and Ghil (1985), Frederiksen and Bell (1987), Frederiksen and Webster (1988), Anderson (1991), Chang and Mak (1993), Frederiksen and Frederiksen (1993a, 1993b), Strong et al. (1993, 1995), Borges and Sardeshmukh (1995), Branstator and Held (1995), de Pondeca et al. (1998) and Li et al. (1999). There have been a number of studies, as well, examining the instability processes involved in the formation of intraseasonal oscillations and equatorial waves (i.e., Frederiksen and Frederiksen 1993a, 1997; Frederiksen 2002, and references therein).

Recently, Frederiksen and Branstator (2001), hereafter FB1) examined how the inclusion of the annual cycle in the basic state of the linear barotropic vorticity equation affects the properties of its leading eigenmodes. They constructed the propagator over a 1-yr period, using a time-dependent basic state taken from monthly averaged 300-hPa reanalyzed observed streamfunction fields linearly interpolated between the different months, and calculated its eigenvalues and eigenvectors, termed finite-time normal modes (FTNMs) by Frederiksen (1997) and Wei and Frederiksen (2004) who examined the roles of FTNMs in error growth during block development. The leading FTNMs of FB1 were found to be large-scale modes with seasonal and intraseasonal variability in their structures, growth rates, and amplitudes. Their fastest-growing FTNM, for example, has its largest growth rate in early northern winter and its largest relative amplitude in boreal spring, when its equatorial penetration is also the largest. The other fast-growing FTNMs also have their largest relative amplitudes in the first half of the year and plummet in late boreal spring and summer. FB1 also found that both intramodal interference effects, between the eastward and westward propagating components of single traveling modes contributing to the fastest-growing FTNM, and intermodal interference effects play significant roles in the evolution of FTNM1, particularly in boreal spring.

The purpose of the present paper is to employ an analogous methodology to that used in FB1 to study the seasonal variability of teleconnection patterns determined from reanalyzed observations and general circulation model simulations. Our study is also complementary to the recent work of Branstator and Frederiksen (2003, hereafter denoted BF03) who focused on other aspects of low-frequency variability of the same datasets. We aim to examine the extent to which finite-time normal mode instability theory is able to provide insights into the structural and amplitude variability of teleconnection patterns as they fluctuate during the annual cycle. In particular, we are interested in discovering whether the distinct seasonal cycle in the growth rates and amplitudes of leading FTNMs, with maximum amplitudes reached in late boreal winter or early spring, and minimum amplitudes occurring in boreal autumn, also carries over to observed and model teleconnection patterns. In FB1 it was also found that leading FTNMs attain growth rates during each season that are generally similar to the growth one would expect from normal modes calculated for that season. We are interested in seeing whether this finding, that perturbations are reacting to the seasonally changing basic state faster than the state is changing, also carries over to teleconnection patterns, since it would explain why linear planetary wave models with time-independent basic states can provide useful insights.

A further aim of this paper is to examine the potential application of our results to the predictability of interannual variability. A number of models of climate prediction over the tropical Pacific Ocean encounter a predictability barrier in boreal spring (Latif and Graham 1991; Cane 1991) when lagged correlations between the monthly mean Southern Oscillation index also decrease rapidly (Webster and Yang 1992; Webster 1995). On the basis of the work in FB1, it seems likely that a contributing cause of the boreal spring predictability and correlation barrier may be the fact that amplitudes of the large-scale instabilities of the atmospheric circulation have peaks during the first half of the year. Here, we aim to examine whether observed teleconnection patterns also have a generally similar seasonal cycle to leading FTNMs (FB1) with maximum equatorial penetration in boreal spring. Such a finding would add weight to the suggested close relationship between the boreal spring predictability barrier and the amplitudes of the large-scale instabilities and teleconnection patterns of the atmospheric circulation.

The methodology that we employ includes and generalizes principal oscillation pattern (POP) analysis (Hasselmann 1988; von Storch et al. 1995) in which the teleconnection patterns are the eigenmodes of an empirical operator determined by long-time 300-hPa streamfunction datasets of atmospheric fluctuations. POP analysis has played a parallel and complementary role in our understanding of the dynamics of atmospheric disturbances to that of normal mode instability theory. As reviewed by von Storch and Zwiers (1999), POP analysis has been widely applied to atmospheric and oceanic variability. Topics covered include teleconnection patterns associated with El Niño–Southern Oscillation (von Storch and Xu 1990; Barnett et al. 1993; Xu 1992), intraseasonal oscillations (von Storch et al. 1988; von Storch and Xu 1990; von Storch and Baumhefner 1991), coupled ocean–atmosphere variability (Blumenthal 1991; Xu 1992), and tropospheric baroclinic waves (Schnur et al. 1993).

In fact, our focus in this paper will primarily be on the generalization to what we term finite-time POPs (FTPOPs) for which the statistics of the fluctuations are cyclostationary (Blumenthal 1991; von Storch et al. 1995), with a period of 1 yr, rather than stationary. The FTPOPs are just the FTNMs of an empirical propagator obtained by fitting a stochastic model to seasonally varying datasets. We also compare the behavior of the FTPOPs with leading monthly POPs and leading monthly empirical orthogonal functions (EOFs) based on streamfunction. The seasonality of geopotential height EOFs for low-frequency atmospheric variability has previously been examined in detail by Horel (1981), Barnston and Livezey (1987), Barnston and Van den Dool (1993), and Wallace et al. (1993). Our discussion of the seasonality of streamfunction EOFs will therefore be rather brief, focusing on streamfunction EOF1, which represents subtropical variability and has no counterpart amongst our leading geopotential height EOFs.

The plan of this paper is as follows: in section 2, we review the theoretical basis of POP analysis, in which a linear stochastic model is fitted to the fluctuations about the mean state. In the case when the dataset is statistically stationary, the eigenvectors of the time-independent stability matrix characterizing the stochastic equation are the usual POPs (Hasselmann 1988). If, instead, the dataset has cyclostationary statistics, with, for example, a period of 1 yr, then the stability matrix is time-dependent and the eigenvectors of the corresponding 1-yr propagator, the FTPOPs, are the analogs of FTNMs. The focus of much of this paper will be on the general properties of FTPOPs, based on twice-daily 300-hPa streamfunction reanalyzed observational fields for the 40-yr period starting on 1 January 1958, and on corresponding daily 497- and 900-yr general circulation model datasets, and how they compare with FTNMs. We also find it instructive to relate FTPOP behavior to the properties of leading monthly POPs and leading monthly streamfunction EOFs, which are discussed in section 3. In section 4, we analyze the growth rates, periods, and time evolutions of some of the leading FTPOPs and relate their structures to leading monthly POPs and monthly EOFs. In section 5 we reapply much of our analysis of EOFs, POPs, and FTPOPs to a 900-yr dataset generated as an ensemble of 20 GCM simulations forced by SST as they were observed to evolve between 1950 and 1994. To examine the effect of interannual variability of SSTs on atmospheric variability, we also perform the same analysis in section 5 for a 497-yr dataset generated by a GCM simulation in which the atmosphere sees the same averaged but seasonally varying SSTs each year. In section 6, we consider the sensitivity of our results to the choice of different datasets. Our conclusions are summarized in section 7. In the appendix, we examine the roles of intramodal and intermodal interference effects in the evolution of FTPOP1.

## 2. Theory

In this section we consider the problem of fitting a linear stochastic model to daily or twice-daily 300-hPa streamfunction fields. We consider the problem when the statistics of the fluctuations about the mean are stationary and when they are cyclostationary (Blumenthal 1991; von Storch et al. 1995; von Storch and Zwiers 1999). We also summarize essential details of the eigenmodes of the stability matrix for stationary statistics and of the propagator for cyclostationary statistics.

### a. Principal oscillation patterns

Suppose we wish to fit a linear stochastic model to a statistically stationary dataset with mean **x** and fluctuations about the mean **x**(*t*). In our case, **x**(*t*) is the column vector of spherical harmonic spectral components truncated rhomboidally at wavenumber-15 and based on either twice-daily National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis 300-hPa streamfunction fields or corresponding daily 497- and 900-yr general circulation model datasets. We focus in this paper on reanalyzed observational data from the 40-yr period starting on 1 January 1958, except in sections 5 and 6 where we also consider datasets obtained from GCM simulations for much longer time periods.

Our linear stochastic model has the form

where 𝗠 is a matrix to be determined from the data, and **f**(*t*) represents noise. Since our data are only sampled every Δ*t* = 12, or 24 h, we estimate the stability matrix 𝗠 through the associated finite-difference equation

where 𝗜 is the unit matrix. The estimate of 𝗠 that minimizes the noise is then given through Gauss' theorem of least squares (Box and Jenkins 1976, their appendix A7.2) as

where + denotes Hermitian conjugate, and angular brackets denote ensemble (or time) means. For a stationary process (*d*/*dt*)〈**x**(*t*)**x**^{+}(*t*)〉 = 0, and hence

In the temporal white noise approximation, 𝗙(*t*) is the covariance matrix of the noise

where *δ*(*t* − *s*) is the Dirac delta function.

The eigenvectors of the empirically determined matrix 𝗠 are POPs (Hasselmann 1988), or empirical normal modes. In this study we shall also consider the eigenvectors of the covariance matrix 〈**x**(*t*)**x**^{+}(*t*)〉, the EOFs.

### b. Finite-time principal oscillation patterns

Suppose now that the mean **x**(*t*) is time dependent and we wish to fit to the data of fluctuations **x**(*t*) a stochastic model

where 𝗠(*t*) is now time dependent, and **f**(*t*) represents noise. We again estimate the matrix 𝗠(*t*) by the right-hand side of Eq. (2.3) with the mean averaged over a time interval centered on *t*. In the case of our long 497- and 900-yr model datasets, the mean **x**(*t*) could be simply estimated as the average, over the years, on a given day *t* of the year. However, on the basis of experimentation we find that for shorter datasets, such as the 40-yr reanalyzed observational dataset, it is preferable to estimate the mean as a combined multiyear and running mean, and that a 10-day running mean is our preferred choice for both observational and model datasets. The solution to Eq. (2.6) may then be written as

where 𝗚(*t, s*) is the propagator that has an integral representation [FB1, their Eq. (2.4)].

The eigenvectors of the propagator 𝗚(*t,* 0) are the natural generalization of POPs to the case of time-dependent statistics. We call the eigenmodes of the empirical propagator finite-time POPs or empirical finite-time normal modes following the terminology of Frederiksen (1997). We use a predictor-corrector scheme to calculate the propagator between successive time steps *t _{k}* and

*t*

_{k}_{−1}as

where *δt*(= *t _{k}* −

*t*

_{k}_{−1}) is a half-hour time step. Then the propagator between

*t*= 0 and

_{o}*t*=

*t*becomes

_{l}The matrix 𝗠(*t*_{k} + *t*_{k−1}/2) is obtained by linear interpolation between successive monthly matrices.

The FTPOPs between an initial time *t* = 0 and a final time *T*, taken here to be 1 yr, are the eigenvectors of the eigenvalue–eigenvector problem

where *λ*^{υ} = *λ*^{υ}_{r} + *i**λ*^{υ}_{i} are the eigenvalues, *ϕ** ^{υ}* are the column eigenvectors of spectral coefficients, and

*N*is the total length of the vectors. As in FB1 we define the global growth rate

*ω*

^{υ}

_{i}and the phase frequency

*ω*

^{υ}

_{r}through the relationship

Thus,

where arctan is a multivalued function whose appropriate branch needs to be chosen. As in the case of FTNMs (FB1), the global growth rate is the average rate at which the corresponding FTPOP grows during a time interval *T*. However, FTPOPs also have local or instantaneous growth rates that are functions of time, and result in bursts of relative growth and decay about the global growth rates as discussed in section 4.

FTPOPs, being just the FTNMs of an empirically determined propagator, again consist of three generic types. “Generalized traveling” FTPOPs have *λ*^{υ}_{i} ≠ 0 and occur in complex conjugate pairs. FTPOPs with *λ*^{υ}_{i} = 0 and *λ*^{υ}_{r} > 0 occur singly and are “recurring” FTPOPs because their structures at *t* = 0 repeat at *t* = *T* (but with different amplitude). FTPOPs with *λ*^{υ}_{i} = 0 and *λ*^{υ}_{r} < 0 also occur singly and are “flip” FTPOPs since their structures at *t* = 0 repeat with opposite sign *t* = *T* (but with different amplitude).

If, as in the case to be considered in this paper, the stability matrix 𝗠(*t*) is periodic, then the linear system satisfies the conditions for Floquet theory (Yakubovich and Starzhinskii 1975; Grimshaw 1990; Strong et al. 1995; FB1and references therein). That is, if

then

where *ρ* is a positive integer. These are also the conditions for cyclostationary POP analysis (Blumenthal 1991; von Storch et al. 1995). In the absence of forcing, Eq. (2.13) implies that as *ρ* → ∞ all initial disturbances (apart from a set of measure zero) converge to 𝗚(*t,* 0)*ϕ*^{1}[*T,* 0], the leading (least damped) FTPOP.

## 3. Fluctuations, EOFs, and POPs for reanalyzed observations

Here we briefly describe the seasonal variability of observed fluctuations about monthly means for a 40-yr dataset of twice-daily 300-hPa streamfunction fields, and we outline some properties of the associated EOFs and POPs. Throughout the study, our source for observed fields is the NCEP–NCAR reanalysis project (Kalnay et al. 1996). And as in all investigations that use such estimates of observed behavior, we must recognize the limitations in accuracy as well as spatial and temporal inhomogeneities inherent in fields derived by combining model output with raw observations of nonuniform quality and coverage. To illustrate the typical large-scale structure of leading EOFs and POPs, we focus on their structures for January. We also examine the properties of the leading EOF and leading POP in March for comparison with the leading FTPOP, which has largest relative amplitude in this month.

### a. Fluctuations

The variability of 300-hPa streamfunction fluctuations based on the twice-daily reanalyzed observational 40-yr dataset follows essentially the same seasonal cycle as that described by BF03 who analyzed monthly averaged NCEP–NCAR reanalysis data for the same period beginning in 1958. The geographical patterns of standard deviations of the anomalies are as shown in Fig. 7 of BF03 for each of the four seasons. The zonally averaged standard deviation of twice-daily streamfunction fluctuations also follows the same seasonality as that shown in Fig. 8 of BF03. In the Northern Hemisphere, the fluctuations have a peak near 50°N in February and drop to almost half this value at the same latitude in July. In the Southern Hemisphere, there is a maximum near 50°S that shows much less seasonality than in the Northern Hemisphere with largest amplitudes in December–May. In June to August, the Southern Hemisphere belt of large amplitudes broadens and extends into both the subtropics and high latitudes.

### b. EOFs

Figure 1 shows the Northern Hemisphere fields of the four leading global EOFs of the 300-hPa streamfunction for January based on the 40-yr dataset. The Pacific–North American pattern is very evident in Fig. 1b which shows EOF2 while the centers of largest amplitude of EOF4 in Fig. 1d coincide with those of the North Atlantic Oscillation (NAO). EOF3 in Fig. 1c has a more zonally symmetric structure with opposite signs in high- and midlatitudes. Essentially the same patterns (but with relatively smaller amplitudes in low latitudes) are also displayed by some of the leading EOFs based on the corresponding 300-hPa geopotential height dataset (not shown). Very similar patterns have previously been obtained in EOF, rotated EOF, and teleconnectivity studies with monthly averaged geopotential height data (e.g., Wallace and Gutzler 1981; Barnston and Livezey 1987).

EOF1, for which the Northern Hemisphere streamfunction field is shown in Fig. 1a, is a large-scale zonally symmetric pattern that captures midlatitude and subtropical variability in both hemispheres. It has no counterpart among our leading geopotential height EOFs and does not appear to have been discussed in other studies of low-frequency variability based on geopotential heights. Branstator (1990, his Fig. 2a) however found that his EOF1 of monthly mean 300-hPa streamfunction generated in a perpetual January simulation with the NCAR Community Climate Model (CCM) had a similar zonally symmetric structure. Zonally symmetric patterns with significant subtropical amplitudes are also evident in the numerical study of Hsu et al. (1990), their Figs. 9 and 10) and in the observational studies of low-frequency variability based on zonal wind EOFs by Schubert and Park (1991, their Fig. 3a), and on streamfunction teleconnectivity by Hsu and Lin (1992, their Fig. 2a).

The subtropical zonally symmetric pattern is, in fact, the dominant streamfunction EOF in each month. Figure 2a shows the Northern Hemisphere streamfunction field of EOF1 for March. Again, the Southern Hemisphere structure of the leading streamfunction EOF in each month (not shown) is largely zonally symmetric and similar to that of FTPOP1 for March shown in Fig. 2d. The EOF1 patterns in each hemisphere are remarkably robust throughout the annual cycle with pattern correlations between EOF1 for January and EOFs1 for the other months greater than or equal to 0.7. In the streamfunction norm, EOF1 also explains the largest percentage of the variance. For example, for January, EOF1 explains 11.7% of the variance and EOF2 to EOF5 explain 4.4%, 3.9%, 3.0%, and 2.8%, respectively, resulting in a total explained variance of 25.8%. In contrast, the corresponding geopotential height EOFs explain 7.3%, 5.6%, 5.1%, 5.0%, and 4.2% of the variance, respectively, and the total variance explained by the first five geopotential height EOFs is 27.2%. As might be expected, because our EOFs are based on global data, the five leading EOFs explain less of the variance than EOFs based on data for a smaller region such as just the Northern Hemisphere (Barnston and Livezey 1987).

### c. POPs

The atmospheric data for a given month (considered over several years) are not statistically stationary but the stability matrix can still be calculated through Eq. (2.3). For the January standard dataset, the five leading POPs have global periods *T _{r}* and global (decay)

*e*-folding times

*τ*of

_{i}days, respectively. By construction they are all damped, so the leading modes are those with longest damping times. Figure 3a shows an example of the structure of POP2 for subsequent comparison with FTPOP1 for January. The imaginary part of POP2 is weak so only the real part is plotted. POP2 has many of the same centers and broadly similar structure to the Pacific–North American (PNA) pattern evident in EOF2 of Fig. 1b; their pattern correlation, taken over both hemispheres, is *A _{c}* = 0.58. In all months, the leading POPs are essentially a superposition of 5 to 10 of the leading EOFs.

The average growth rates of the five leading least-damped POPs are shown in Fig. 4 for each month. We note that growth rates are largest in northern winter and generally smallest in northern summer. In these respects, they are similar to the average growth rates of the normal modes growing on monthly averaged basic states for the period between 1958 and 1997 shown in Fig. 9a of FB1.

For a statistically steady dataset, the decay rates of all the POPs are balanced on average by the injection of energy by the random forcing. BF03 for example, discusses in detail the role of random forcing in maintaining the amplitudes of fluctuations in damped barotropic simulations. For January, the standard deviation of the random forcing (not shown) obtained through Eqs. (2.4) and (2.5) exhibits the characteristic storm tracks in both the Northern (Blackmon 1976) and Southern (Trenberth 1982) Hemispheres indicating that synoptic-scale disturbances are major forcing mechanisms for the large-scale low-frequency disturbances. This conclusion also holds for all other months with the structure of the standard deviation of the random forcing following the seasonal cycle of storm track activity in both hemispheres (not shown).

## 4. FTPOPs for reanalyzed observations

In FB1 we found it useful to understand the behavior of leading FTNMs in terms of the fast-growing normal modes for corresponding months. In particular, the roles of intramodal growth and intramodal and intermodal interference effects in the growth cycle of leading FTNMs were considered in detail. Since FTPOPs are just the FTNMs of an empirical propagator, one might expect that there are many qualitative similarities in the behaviors of FTPOPs and FTNMs. This is indeed the case and our discussion of the relationships between FTPOPs and POPs will therefore be much briefer. In this section we also examine relationships between FTPOPs and EOFs. In our study of FTPOPs, we have found it convenient to divide the annual cycle *T* into 360 effective days with each month consisting of 30 effective days. Comparisons with POPs and EOFs for monthly averaged basic states are therefore made with FTPOPs on the 15th of the corresponding month.

### a. Growth rates, e-folding times, and periods

The eigenvalues *λ*^{υ} = *λ*^{υ}_{r} + *i**λ*^{υ}_{i}, mode type for the five dominant FTPOPs, as well the corresponding global growth rates, *e*-folding times- and periods are shown in Table 1. As in the case of FTNMs of FB1, the amplification factors |*λ ^{υ}*| give the average amplification (in the case of FTPOPs decay) of the FTPOPs over a 1-yr period. We note that the five least-damped modes have reasonable lifetimes with

*e*-folding (decay) times ranging from −23.6 to −14.6 days. Periods are 1 or 2 yr, corresponding to recurring or flip modes, apart from mode 5, which is a generalized traveling mode. As discussed in FB1 and in section 2, the decay of an FTPOP is not uniform but varies with time in the manner detailed below.

In FB1 it was found that FTNMs, and by implication FTPOPs, have local growth rates and relative amplification factors that vary with time. To define these measures of local growth we first let

represent the initial column vector based on the *υ*th eigenfunction spherical harmonic spectral coefficient ϕ^{υ}_{mn} [Eq. (A.2) of FB1], where *m* denotes zonal wavenumber, and *n* total wavenumber. Here, T denotes transpose. We also define the streamfunction *L*_{2} norm of any vector **x** by

The evolved column vector of a FTPOP is then given by

and the total amplification factor of the disturbance between 0 and *t* is

We may then define the local total growth rate by

where *ω*^{υ}_{i} is the global growth rate and ω̂^{υ}_{i}(*t*) is the local relative growth rate which gives the departures from *ω*^{υ}_{i}. The local relative growth rate is related to *R ^{υ}*(

*t*), the relative amplification factor through

where

As noted in FB1, *R*^{υ}(*t*) can be viewed as an integral measure of *ω̂*^{υ}_{i}(*t*).

Figure 5a shows the local total growth rate, *ω̂*^{1}_{i}(*t*) (dashed) and *R*^{1}(*t*) (thick solid) for FTPOP1. Comparing this diagram with the corresponding results in Fig. 4a of FB1 for FTNM1, we note a number of general similarities. First, the maximum relative amplification factor occurs in early boreal spring and has a magnitude of around twice the value in January. It then plummets in late boreal spring, attaining low values in boreal summer and autumn, and with generally increasing values in late autumn and early winter. We also note that the minimum in *R*^{1}(*t*) for both FTPOP1 and FTNM1 occurs between October and November. These similarities are also reflected in the local growth rates. For example, for both FTNM1 and FTPOP1, the maximum in *R*^{1}(*t*) in early boreal spring is preceded, by a few weeks, by a local maximum in the relative growth rate.

Again, the characteristics of FTPOP1 shown in Fig. 5a, including the large amplitudes of the relative amplification factor in the first half of the year, the small values in late boreal summer and throughout autumn, and the increases in boreal winter, are representative of the class of leading FTPOPs. In Fig. 5b, we show the average local total growth rate of the five leading FTPOPs (dashed) and the relative amplification factor *R*(*t*) (thick solid). The amplification factor again has the largest values in the first half of the year, as for the five leading FTNMs in Fig. 4b of FB1, decreases rapidly in late boreal spring and summer and then increases gradually in boreal autumn and winter. The growth rates tend to be smallest on average in northern summer, and largest in northern autumn and winter. This is perhaps more clearly seen in Fig. 4 which shows the 30-day running means of the average local total growth rate of the five leading FTPOPs. The seasonality of the relative amplification factor in Fig. 5b, which reflects the seasonality of the growth rates, is also found in averages taken over the 10 leading FTPOPs. The seasonality of the 30-day running mean of the average growth rates in Fig. 4 with largest values in boreal winter and smallest values in boreal summer is similar to seasonality in the strength of the Northern Hemisphere basic-state flow measured by the scaling coefficient *c _{j}* in Table 1 of FB1. The scaling coefficient

*c*in a given month

_{j}*j*is obtained from a least squares fit of the monthly averaged Northern Hemisphere 300-hPa streamfunction onto the corresponding annual mean Northern Hemisphere basic-state streamfunction. Again there are also local variations due to intramodal and intermodal growth and interference as discussed in FB1 and in the appendix. We note, however, that both the average FTPOP and POP growth rates peak in December and February while for the normal modes and FTNMs in Fig. 9 of FB1 the maxima are in November and December.

### b. Structure and evolution of FTPOP disturbances

Here, we briefly discuss the structures of leading FTPOPs focusing on FTPOP1 and its evolution over the annual cycle. We relate the evolving FTPOP structure to POPs and EOFs. Figure 3b shows the 300-hPa disturbance streamfunction for FTPOP1 on 15 January on the Northern Hemisphere stereographic projection. FTPOP1 displays many of the features of EOF2 for January in Fig. 1b including the distinct Pacific–North American pattern. The global pattern correlation between FTPOP1 and EOF2 in January is *A _{c}* = 0.56 while, of the January POPs, POP2 (Fig. 3a) has largest pattern correlation of

*A*= 0.49 with FTPOP1. By 15 March, FTPOP1 has attained the more zonally symmetric structure shown in Figs. 2c and 2d for the Northern and Southern Hemispheres. At this stage it has the largest pattern correlations with EOF1 for March (Fig. 2a) with

_{c}*A*= 0.84 and POP1 for March (Fig. 2b) with

_{c}*A*= 0.90.

_{c}As was the case for the relationship between normal modes and FTNMs in the study of FB1, the structures of FTPOPs often resemble one of the leading POPs (or leading EOFs) in corresponding months. This similarity may be quantified by calculating pattern correlations between FTPOP1 on the 15th of each month and each of the 10 leading EOFs and POPs for the month. Figure 6 displays these correlations and EOF and POP indices when the correlation is maximized over the two signs of the 10 EOFs and over all phases of the 10 POPs. We see that in most months FTPOP1 has significant correlations with some of the 10 leading EOFs and POPs. Figure 6 also shows the pattern correlation between FTPOP1 and that linear combination of 10 leading EOFs or 10 leading POPs that best fits FTPOP1. We note that while both classes of patterns fit FTPOP1 well, FTPOP1 conforms to the POPs more closely than to the EOFs. We also see that FTPOP1 frequently, but not always, has largest pattern correlation with EOF1 or POP1.

The structural and amplitude changes of the leading FTPOPs as they evolve (not shown) have a similar complexity to that shown in Fig. 5 of FB1 for their FTNM1. The FTPOPs however tend to have larger relative amplitudes in the subtropical regions and in the Southern Hemisphere than do the FTNMs. An example of this is shown in Figs. 2c and 2d for FTPOP1 in March. This may also be seen by comparing Fig. 7a with Fig. 7b of FB1. Our Fig. 7a shows a latitude–time cross section of the zonal average of the absolute value of the 300-hPa streamfunction for FTPOP1 (scaled by exp *ω*^{1}_{i}*t*). We note the fairly comparable amplitudes in the Northern and Southern Hemispheres, particularly during the first half of the year, and the peak amplitudes in March when the equatorial penetration of the disturbance is also the largest. In contrast for FTNM1, in Fig. 7b of FB1, the Southern Hemisphere amplitudes are much smaller than Northern Hemisphere amplitudes. The other four of the five leading FTPOPs also have the largest amplitudes and equatorial penetration during the first half of the year, as shown in Fig. 7b for FTPOP2, and have weaker amplitudes during the rest of the year than FTPOP1. As was the case with FTNMs in FB1, the leading FTPOPs also exhibit intraseasonal variability (not shown) on a similar timescale to that displayed in Fig. 7a of FB1.

## 5. Seasonal variability in GCM experiments with observed SSTs

Next, we perform a similar analysis of atmospheric variability in terms of EOFs, POPs, and particularly FTPOPs for a 900-yr dataset generated as an ensemble of 20 GCM simulations forced by the observed SSTs that evolved between 1950 and 1994. The simulations were performed with version 3 of the NCAR CCM (CCM3) which is described by Kiehl et al. (1998). Some complementary aspects of the GCM climate were examined by Hurrell et al. (1998), Branstator (2002) and by BF03. We call this dataset the *variable SST* dataset and contrast it with a *same SST* dataset of 497 yr of integration of the same model but using the same seasonally varying observed average SSTs each year.

### a. Fluctuations and EOFs

The 300-hPa streamfunction fluctuations based on the daily 900-yr variable SST dataset has essentially the same seasonal variability as that described by BF03 who analyzed a very similar but monthly averaged GCM dataset forced by the observed SSTs for 1950 to 1994. Figures 11 and 12 of BF03 show the internal and external components, respectively, of GCM streamfunction variability at 300-hPa in January, April, June, and October.

For the variable and same SST datasets, the five leading global EOFs for January are quite similar with pattern correlations close to or greater than 0.9. As shown in Table 2 the three leading EOFs for the two GCM datasets also have pattern correlations with the three leading EOF of the 40-yr observed dataset ranging from larger than 0.6 to larger than 0.9.

### b. POPs and FTPOPs

In each month there are leading POPs and leading EOFs that have very similar structures to leading FTPOPs of the model datasets. Figure 8 shows the pattern correlations between FTPOP1 for the 900-yr variable SST dataset and the corresponding EOFs and POPs for which the correlation is a maximum. For example, we note that for January EOF1 and POP3 are very similar in structure to FTPOP1. Again the correlation is maximized over the two signs of the EOFs and over the phases of the POPs. Figure 8 also shows the correlation between FTPOP1 and the best fit of FTPOP1 in terms of the 10 leading EOF or 10 leading POPs. It is clear that the 10 leading POPs provide an excellent fit to FTPOP1 in each month and that a representation in terms of 10 EOFs is also quite good in most months.

#### 1) Global growth rates, *e*-folding times, and periods

Table 3 shows the eigenvalues *λ*^{υ} = *λ*^{υ}_{r} + *i**λ*^{υ}_{i}, growth rates, *e*-folding times, periods, and mode type for the five dominant FTPOPs for the 900-yr variable SST dataset. The *e*-folding (decay) times, which range from −32.5 to −17.7 days, are comparable with those for the leading FTPOPs for the observations (Table 1) but with somewhat larger magnitudes. Table 4 shows the corresponding results for the 497-yr same SST dataset. Here the *e*-folding times range from −20.2 to −14.0 days and are very comparable to those for the FTPOPs for the observations in Table 1. We shall explore the reasons for the differences and similarities in FTPOP *e*-folding times associated with the reanalyzed observed and two GCM datasets below.

#### 2) Structure of FTPOPs

Figures 9a and 9b show the 300-hPa disturbance streamfunction for FTPOP1 on 15 January on the Northern Hemisphere stereographic projection and for the 900-yr variable SST and 497-yr same SST datasets, respectively. We note that FTPOP1 for the variable SST data is a largely zonally symmetric mode somewhat akin to EOF1 for the observations shown in Fig. 1a and to EOF1 for the variable SST data (not shown) with which it has a global pattern correlation of 0.83. In contrast, FTPOP1 for the same SST dataset has a more distinct wave structure with major centers in similar locations to FTPOP1 for the observations shown in Fig. 3b. In the Northern Hemisphere FTPOP1 for the same SST dataset is also quite similar in structure to FTPOP3 for the variable SST data with a Northern Hemisphere pattern correlation of 0.82. We also note that both of these FTPOPs are flip modes with very similar *e*-folding times of −20.2 (FTPOP1 for same SST data) and −20.4 days (FTPOP3 for variable SST data). It appears that the two leading FTPOPs for the 900-yr variable SST data are a reflection of the additional variability induced in the atmosphere by the interannual variability of the SSTs.

#### 3) Local growth, relative amplification, and evolution of FTPOPs

In Figs. 10a and 10b we show the average local total growth rate of the five leading FTPOPs (dashed) and the relative amplification factor *R*(*t*) (solid) for the 900-yr variable SST data and for the 497-yr same SST data, respectively. In both cases, the amplification factors again have largest values in early boreal spring, decrease rapidly in mid-to-late boreal spring, and also show a gradual increase in boreal autumn and winter. In these respects, the behavior is similar to that of corresponding results for FTPOPs based on observations (Fig. 5b) and for the FTNMs shown in Fig. 4b of FB1. The amplification factors in Figs. 10a and 10b for the GCM datasets, however, have the additional feature of secondary maxima in early boreal summer. These, in turn, are associated with subsidiary maxima in the growth rates in late boreal spring. The general character and differences in the growth rates between the FTPOPs for the variable SST data and the same SST data is perhaps more clearly seen in Fig. 10c which shows 30-day running means of the average local total growth rate of the five leading FTPOPs in both cases. The less-negative growth rates for the FTPOPs for the variable SST data compared with the same SST data are clearly evident. Again, growth rates are larger in boreal winter in both datasets with subsidiary maxima in late boreal spring in contrast with results for FTPOPs based on observations (Fig. 4) and for the FTNMs in Fig. 9b of FB1.

A useful way to depict the general character of the structural and amplitude changes of FTPOP1, for example, as it evolves is through a latitude–time cross section of the zonal average of the absolute value of its 300-hPa streamfunction (scaled by exp *ω*^{1}_{i}*t*). Figures 11a and 11b show such Hovmoeller diagrams for FTPOP1 for the variable and same SST data, respectively. We note the very different behavior of FTPOP1 for the variable SST data compared with that for the same SST data. FTPOP1 for the 900-yr data in Fig. 11a has large amplitudes in the subtropical regions in boreal winter and spring, and secondary maxima there and at high southern latitudes in early boreal summer. Its structure should be compared with Fig. 7a which shows the same results for FTPOP1 as for the observations. In contrast, FTPOP1 for the 497-yr same SST data has largest amplitude at high northern latitudes in boreal spring and much weaker amplitudes in the subtropics. In these general respects it bears similarities to the corresponding Hovmoeller diagram for FTNM1 in Fig. 7b of FB1, although Southern Hemisphere amplitudes are clearly larger in Fig. 11b.

These results suggest that for the same SST dataset the contribution of internal instability and variability may be relatively more important while for FTPOPs 1 and 2 for the variable SST dataset forced variability by interannual variations in SSTs may play an important role. The larger growth rates associated with the variable SST dataset compared with observations also suggest that CCM3 may be somewhat more sensitive to interannual variability in SST forcing compared with the atmosphere, though it is less sensitive than several other models (Shukla et al. 2000).

### c. FTNMs

We have also computed the FTNMs of the barotropic vorticity equation with 300-hPa basic states for both the 900-yr variable SST and 497-yr same SST data using the method described in FB1(and with zero drag and an outer rhomboid Laplacian dissipation with viscosity coefficient of 2.5 × 10^{5} m^{2} s^{−1}). Figure 12a shows the average local total growth rate of the five leading FTNMs (dashed) and the relative amplification factor *R*(*t*) (solid) for the 900-yr variable SST data. The corresponding results for the 497-yr same SST data are very similar (not shown); the degree of similarity is perhaps best seen from Fig. 12b which shows 30-day running means of the average local total growth rate of the five leading FTNMs in both cases. These results confirm that the internal instability of the basic states for the variable and same SST data is essentially the same, and the differences in the character and growth of the FTPOPs is due to the different variability of the SST.

The cycle of relative amplification and decay in Fig. 12a is quite similar to that depicted in Fig. 4b of FB1 for the five leading FTNMs based on observations, although for the GCM results, the boreal winter growth occurs slightly later. This is perhaps most evident by comparing Fig. 12b with Fig. 9b of FB1, which shows the corresponding 30-day running means of average local total growth rate of the five leading FTNMs based on observations.

## 6. Sensitivity studies

Eigenvalues and eigenvectors may be sensitive to perturbations to the matrices defining the linear problems of interest (Golub and van Loan 1996). This is the case for theoretical instability problems (Simmons et al. 1983; Anderson 1991), for EOF analyses (Wallace and Gutzler 1981; Barnston and Livezey 1987), and for the empirical normal modes of POP analyses (von Storch et al. 1995). Despite this sensitivity, FB1 found that their general conclusions on the seasonality of FTNMs were robust for basic states based on other long-time averaged datasets. Here, we briefly consider the sensitivity of EOFs and FTPOPs.

### a. EOFs

Barnston and Livezey (1987) made a detailed analysis of the robustness of both rotated and unrotated eigenvectors of the covariance matrix (EOFs or principal components). We shall therefore make our discussion very brief focusing on the leading EOFs for January. The patterns shown in Figs. 1 and 2 are also clearly identifiable in corresponding EOFs based on 7-, 10-, and 30-day running means of the 40-yr reanalyzed observational dataset. For example, global pattern correlations are 0.99, 0.88, 0.89, 0.58, and 0.69, respectively, between the corresponding leading EOFs 1 to 5, for the standard 12-hourly and 30-day running mean data. These results support and add to the findings of Barnston and Livezey (1987) who found that almost all their patterns classified on the basis of monthly mean data were also produced in analyses based on 10-day mean analyses (and 3-month mean analyses) for the months examined (January, April, July, and October).

EOFs, POPs, and FTPOPs also depend on the particular properties of the sampled dataset that they attempt to represent and may not be exactly the same as for a longer statistically stationary or cyclostationary dataset. For example, in Table 5 we show the pattern correlations between the five leading January EOFs for the 900-yr variable SST and corresponding EOFs for the first 450-yr period, and the first and second 45-yr periods of this dataset. On the basis of these results we expect that a 900- or 450-yr dataset is sufficient to accurately define at least the five leading EOFs while for the shorter 45-yr dataset the particular sampling period may significantly influence the structures of all but the leading two EOFs.

### b. FTPOPs

As noted in earlier sections, the leading FTPOPs are essentially a linear combination of 5 to 10 leading EOFs, so we might expect sampling would also affect these empirical modes. Here, we examine whether the general findings regarding the seasonality of FTPOP amplitudes outlined in sections 4 and 5 also apply for other long-time datasets.

Table 6 shows the eigenvalues *λ*^{υ} = *λ*^{υ}_{r} + *i**λ*^{υ}_{i}, growth rates, *e*-folding times, periods, and mode type for the five dominant FTPOPs for the first 450-yr period of the 900-yr variable SST dataset. In all cases, the *e*-folding times of the five leading FTPOPs differ from those of the corresponding 900-yr data by less than 5%. In the case of FTPOPs 4 and 5, for which the *e*-folding times are very close, there is a swapping of orders between the first 450- and 900-yr periods. For the first 450-yr period and the 900-yr period, FTPOPs1 all have relative amplification factors with maxima in February of circa 1.1, minima in November of about 0.35, secondary minima in April of about 0.8, and secondary maxima in June of about 0.95. Again, the average local total growth rate of the five leading FTPOPs for the first 450-yr period, and the 30-day running means, follow annual cycles broadly similar to those for the 900-yr data in Figs. 10b and 10c; peak growth rates occur in December, and primary minima in September with secondary maxima in May to June, and secondary minima in April. So our GCM experiments appear to be long enough to provide robust leading FTPOPs. For shorter 45-yr datasets the particular sampling period significantly influences the seasonality of the leading FTPOPs; the timing of the general features of the boreal cold season maximum growth rate and the minimum at the end of the boreal warm season can differ by as much as two and a half months. As we have noted in earlier sections, there are broad similarities between average amplitudes of relative amplification factors of leading FTPOPs for long-time GCM datasets and for the 40-yr reanalyzed observational dataset.

## 7. Discussion and conclusions

The seasonal variability of 300-hPa atmospheric streamfunction fluctuations has been examined for both reanalyzed observed and general circulation model datasets forced by seasonal variations in sea surface temperatures (SSTs). We have studied the large-scale properties of these fluctuations through their representation in terms of leading empirical orthogonal functions (EOFs), principal oscillation patterns (POPs), and finite-time principal oscillation patterns (FTPOPs). In the case of the model datasets, we have examined the extent to which our findings depend on whether the model atmosphere is forced by interannual variations in SSTs, as in a 900-yr variable SST dataset, or feels the same seasonal variations in SSTs every year as in a 497-yr same SST dataset. A particular aim has been to examine the extent to which the general principles concerning the seasonality of FTNM disturbances encountering the annual cycle of variations in the basic state, established in FB1, also apply to FTPOPs. The FTNMs are the eigenmodes of the propagator, covering the full annual cycle of observed basic states in the barotropic vorticity equation, while the FTPOPs are the eigenmodes of the corresponding propagator derived from fitting a linear stochastic model to the observed atmospheric fluctuations as they also vary with the annual cycle.

We have found that in many respects, the properties of theoretical FTNMs established in FB1 do carry over to their observational and GCM-modeled counterparts, the FTPOPs. The most striking similarity between FTNMs and FTPOPs is in the seasonality of the growth rates of the leading modes. In both theoretical and empirical settings there is a distinct annual cycle of these rates with the maximum occurring during the middle of the boreal cold season and a broad minimum being present during the boreal warm season. The amplitude of this annual cycle is also rather similar with the leading five FTPOPs having a difference in average growth rates of about 0.02 day^{−1} between warm and cold seasons (Fig. 4), while the leading FTNMs have a corresponding range of roughly twice this amount, depending on details of how one treats of the seasonality of the basic state in the calculation (Fig. 9 of FB1). We note that the leading FTNMs of FB1 were analyzed in the case of zero drag and were interpreted as growing instabilities while FTPOPs, as the statistical modes of a stationary process, all decay. However, as discussed by BF03, if the nonlinear terms are approximated by drag and random noise then the linear problem also gives insight into stationary processes. In particular, the drag term just reduces growth rates (to below neutrality), and has no effect on frequencies or modal structures (FB1) and the FTNMs are the FTPOPs of the cyclostationary process for which the nonlinear terms are approximated by drag and random noise.

The similarity in the seasonality of growth characteristics is even more evident if one considers the time-integrated effects of growth, as given by our relative amplification rate. In this case, one finds that for both theoretical and empirical modes, maximum amplitudes are reached near the end of March and minimum amplitudes occur in early November. It is not surprising that it is in integrated quantities that theory and observations coincide best; as FB1 pointed out FTNMs (and FTPOPs; appendix) can have bursts of growth and decay for short periods through intramodal and intermodal interference effects. These come about from interference between eastward and westward components of a single propagating wave and between several normal modes of the system, respectively. One cannot expect theory and observations to agree on the exact timing of such processes during the annual cycle. On the other hand, the overall change in the large-scale gradients of the time mean circulation and its similar influence on the growth characteristics of theoretical FTNMs and empirical FTPOPs should show up in time- and mode-averaged quantities, and is likely to be the reason for the similarities in amplification we just noted. Support for this explanation comes from the similarity we noted between mode-average growth rates and the seasonality of time mean basic-state strength, as given by the *c _{j}* parameter alluded to in section 4a and quantified in FB1.

One further similarity we have found between growth properties of leading FTNMs and FTPOPs is that both attain growth rates during each season that are similar to the growth one would expect from normal modes and POPs calculated for that season. This means that in both cases, perturbations are reacting to the seasonally changing basic state faster than the state is changing. In FB1 it was shown that this is only a first approximation, particularly during spring when intermodal growth appears to make a significant contribution, but it serves to explain why linear planetary wave models that employ basic states that are constant in time can be useful.

When considering the structure of FTPOPs and FTNMs, finding a correspondence between empirical behavior and theory is not as straightforward as for growth rates. Aside from the fact that the leading observational FTPOP and FTNM are composed of large-scale anomalies, even some of the grossest features of the two do not agree. In particular, while FTNM1 is largely confined to the Northern Hemisphere and extends into the Tropics and subtropics only during boreal winter and spring, observational FTPOP1 has large amplitudes throughout the globe during all seasons. Repeating our FTPOP analysis for data generated by the two GCM experiments helped resolve this apparent discrepancy. When we used data from the variable SST experiment, FTPOPs matched observed behavior in every salient respect. The seasonal cycle of growth rates for the leading FTPOPs was similar and the structure of FTPOP1 (and FTPOP2) again disagreed with FTNM1 by being strong throughout the globe during all seasons. On the other hand, when we used data from the same SST experiment, though there were only minor changes in the growth rates, the structure of FTPOP1 became much more like FTNM1. In this experiment, FTPOP1 has higher amplitude in the Northern Hemisphere than in the Southern Hemisphere and there is a distinct seasonal cycle with large Northern Hemisphere amplitudes being confined to high latitudes in summer and early fall and stretching into the subtropics other times of year. From this comparison we concluded that the leading pair of FTPOPs in GCMs with interannually varying SSTs (and in nature) attain their prominence not only from the effects of internal atmospheric processes but also reflect prominent, recurring effects of forcing from the ocean. It is for this reason that they do not match the structure of the leading FTNMs. By contrast, the leading structures produced without the influence of interannual variability (as well as slightly less prominent structures, as exemplified by FTPOP3 in the variable SSTs) do match many attributes of the structure of the leading FTNMs and thus are largely understood solely in terms of internal dynamical processes.

Taken together, then, the analysis of the seasonal cycle of FTPOP growth and structure are complementary to the results of BF03. That study found that the seasonal cycle of variance and the structure of interannual variability could be largely understood in terms of dynamical effects of the seasonally varying time-averaged circulation. Here, by focusing on the structures of those features singled out by the leading two FTPOPs, we also see the influence of the time-varying circulation, but we also can detect an additional measure of organization coming from the influence of interannual variability of the oceans. In this way FB1, BF03 and the present study unite past investigations that have either emphasized the role of internal atmospheric dynamics or the role of SST anomalies in organizing interannual atmospheric variability.

A second application of our results concerns the predictability of interannual variability. In FB1 we suggested that a contributing cause of the boreal spring predictability and correlation barrier may be the fact that amplitudes of the large-scale instabilities of the atmospheric circulation have peaks during the first half of the year. Some models of climate prediction over the tropical Pacific Ocean encounter a predictability barrier in boreal spring (Latif and Graham 1991; Cane 1991) when lagged correlations between the monthly mean Southern Oscillation index also decrease rapidly (Webster and Yang 1992; Webster 1995). Here we have found that large-scale coherent structures, represented by leading FTPOP teleconnection patterns, have a generally similar seasonal cycle of relative amplitude increases and decreases to that of the leading FTNM instabilities of FB1. In fact, for both the 40-yr observed and 900-yr variable SST datasets, the equatorial penetration of FTPOP1 during its maximum in boreal spring is more dramatic and occurs in both hemispheres. These results again suggest a close relationship between the boreal spring predictability barrier and the amplitudes of the large-scale instabilities and teleconnection patterns of the atmospheric circulation.

## Acknowledgments

It is a pleasure to thank Steve Kepert and Andy Mai for assistance with this work.

## REFERENCES

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### APPENDIX

#### Intramodal and Intermodal Interference Effects

In FB1 we found that to a first approximation, the dramatic amplification and decay of FTNM1 during boreal spring could be explained by intramodal growth and interference effects. In March and April, FTNM1 essentially followed the local relative growth and decay cycle of normal mode 1 for 15 March. Here we briefly note that a similar relationship exists between the behavior of FTPOPs and POPs.

For a POP, the relative amplification factor and local total growth rate are again defined in Eqs. (4.7) and (4.5), respectively, but with the propagator 𝗚(*t,* 0) = exp(𝗠*t*) where 𝗠 is the time-independent stability matrix for the month. For POP1 for March, *R*^{1}(*t*) and the relative local growth rate have essentially the same variability as shown for mode 1 for the 15 March basic state in Fig. 3 of FB1. Again, *R*^{1}(*t*) increases from 1.0 at phase 0° (day 0) to 2.05 at phase 90° (day 73), compared with the value of 4.04 for mode 1 for 15 March, and then decreases back to unity at phase 180° (day 146). The period of POP1 of 292 days is also very similar to that for mode 1 for 15 March (290 days).

During its evolution the correlation of FTPOP1 with POP1 for March increases from 0.438 on 11 February to a maximum of 0.915 on 21 March and then decreases to 0.618 on 3 May. As suspected, to a first approximation, the evolution of FTPOP1 during boreal spring is essentially that of POP1 for March during its first half-cycle of growth and decay related to intramodal interference effect. Nevertheless, as was the case for FTNM1 of FB1 intermodal interference effects also play a significant role during the evolution of FTPOP1; that is, there are significant contributions from other March POPs to FTPOP1 during March.

## Footnotes

*Corresponding author address:* Dr. Jorgen S. Frederiksen, CSIRO Atmospheric Research, 170-121 Station Street, PMB#1, Aspendale 3195, Victoria, Australia. Email: jorgen.frederiksen@csiro.au