The barotropic instability of time-dependent observed basic states that are periodic, with a period of 1 yr covering the complete annual cycle, is analyzed using Floquet theory. The time-dependent basic state is constructed from observed monthly averaged 300-mb streamfunction fields linearly interpolated between the different months. The propagator over the 1-yr period is constructed, and its eigenvalues and some of the fastest-growing eigenvectors, termed finite-time normal modes (FTNMs), are calculated. The fast-growing FTNMs are large-scale modes with generally largest amplitudes in the Northern Hemisphere. They exhibit intraseasonal variability in their structures, as well as longer period variations, and their amplification rates vary with time. The fastest-growing FTNM has its largest growth rate in early northern winter and its amplification has maximum cumulative effect in boreal spring when the equatorward penetration of this disturbance is also the largest. The other fast-growing FTNMs also have largest amplitudes during the first half of the year.
In all months, there are fast-growing normal modes of the monthly averaged stationary basic states that have large pattern correlations with the fastest-growing FTNM for the time-dependent basic state. For some months the individual normal modes experience dramatic local variations in growth rate; these bursts of relative growth and decay are associated with intramodal interference effects between the eastward and westward propagating components of a single traveling normal mode. Both intramodal and intermodal interference effects play significant roles in the evolution of the fastest-growing FTNM, particularly in boreal spring.
The behavior of FTNM instabilities is also examined in simplified situations including a semianalytical Floquet model in which the space and time dependencies of the stability matrix are separable. In this model, temporal variations in growth rates are directly linked to seasonality in the intensity of the climatological state.
It has long been recognized that the horizontal and vertical variations in the mean atmospheric circulation play major roles in the locations and structures of large-scale atmospheric disturbances. The barotropic study of Simmons et al. (1983) and the barotropic and baroclinic study of Frederiksen (1983) showed that perturbation growth resulting from these variations could lead to disturbances structurally similar to prominent observed low-frequency anomalies. Indeed, their dominant barotropic and equivalent barotropic modes exhibit remarkable similarities to the observed Pacific–North American, east Atlantic, and North Atlantic Oscillation teleconnection patterns (Wallace and Gutzler 1981; van Loon and Rogers 1978). As pointed out in a number of early studies (Simmons et al. 1983; Frederiksen 1983; Dole 1986; Frederiksen and Webster 1988) and more recently by Borges and Sardeshmukh (1995) and Newman et al. (1997) barotropic instability alone is unlikely to produce sufficient growth to explain observed low-frequency amplitudes. But normal-mode analysis is a useful way of determining structures that can grow at the expense of the time mean state and thus have a dynamical advantage in the natural selection process. This realization has motivated a number of subsequent studies of large-scale normal modes and their roles in the formation of teleconnection patterns and intraseasonal oscillations including the works by Branstator (1985, 1987), Legras and Ghil (1985), Anderson (1991), Frederiksen and Frederiksen (1993), Strong et al. (1993), and Branstator and Held (1995). More detailed reviews of the literature are given in Frederiksen and Webster (1988) and Frederiksen (1997).
The above studies have considered stationary basic states, but there are good reasons to wonder whether time-dependent basic states might be more appropriate for many situations. It has turned out that the leading barotropic modes in these studies typically have natural periods ranging anywhere from about a month to infinity. These modes are usually interpreted as being theoretical counterparts to observed circulation perturbations that last as much as a season, if not longer. Therefore, the basic state that one should use in such a study must be one that is appropriate for several months, suggesting that rather than a time-invariant state, perhaps a state that includes the month-to-month variation in the climatological state might be more appropriate. With this in mind, the aim of the study described in this paper is to determine if and how the inclusion of an annual cycle in the basic state of the linear nondivergent barotropic vorticity equation, the model used in many related investigations, changes the characteristics of its leading eigenmodes.
From Floquet theory (Yakubovick and Starzhinskii 1975; Grimshaw 1990; Iooss and Joseph 1990) we know that significant time variations in the basic state of a system can have a major impact on the system’s modes. A well-known example would be the trapeze instability (Orlanski 1973). Thus, if seasonality of the time mean state is large enough, there is the possibility that it could produce new modes of instability not found in previous studies with time-invariant basic states. These could be in addition to, or in place of the modes that occur with time-independent basic states. Moreover, we know from previous studies with time-invariant basic states that seasonal changes in the time mean state are large enough to be dynamically significant. After all, Simmons et al. (1983) found that the leading modes of the barotropic vorticity equation were completely different in structure and frequency for Northern and Southern Hemisphere January mean states, a contrast that is qualitatively similar to the difference between winter and summer mean conditions in the Northern Hemisphere. Furthermore, other studies, though not employing eigenanalysis, have demonstrated a dependence of the dynamical characteristics of the barotropic linear vorticity equation, depending on the seasonal mean used as a basic state. For example, Branstator (1999, manuscript submitted to J. Climate) has found that seasonality in the time mean state is largely responsible for the seasonal swings in the distribution and structure of interannual variability observed in nature (Barnston and van den Dool 1993; Branstator and Frederiksen 1999, manuscript submitted to J. Climate). And Newman and Sardeshmukh (1998) have shown that seasonal changes in the time mean are large enough to have significant influence on the energy propagation characteristic of this same model.
In spite of their potential impact on low-frequency modes, few studies have considered time-varying basic states in a low-frequency context. Methods have been developed for investigating the effect of time-varying basic states and applied to shorter timescale perturbations. Singular vectors growing on time-dependent basic states are now commonly used in studies of synoptic error growth (Molteni and Palmer 1993; Frederiksen 2000 presents a detailed literature review) and data assimilation. Frederiksen (1997, 2000) examined the finite-time instability of observed four-dimensional space–time basic states during cases of block development using a construct he called a finite-time normal mode (FTNM). One example of a study that has considered longer timescale perturbations is that of Strong et al. (1995), who examined the barotropic instability of time-dependent idealized basic states taken from barotropic model integrations incorporating an annual cycle. They performed an eigenanalysis of the periodic basic state based on Floquet theory. They found that the fastest-growing eigenmode was an intraseasonal oscillation in the extratropics with a period near 40 days; it was modulated by the annual cycle to have largest amplitude in boreal winter.
To address our goal of studying the effect of the annual cycle on modes of the barotropic vorticity equation, we employ the FTNM method and an observed basic state based on the annual cycle of climatological average 300-mb streamfunction. In section 2, we give details of the barotropic model, linearized about a time-dependent basic state, which is used in this study. We describe how we construct the propagator covering the one-year period and calculate its eigenvalues and eigenvectors, which are the FTNMs. In this section we also describe how the annual cycle basic state is constructed from climatological monthly means and we give a brief summary of Floquet theory for periodic basic states. Section 3 examines the leading normal modes of each of the monthly mean climatological states used to construct the annual cycle basic state. This analysis documents the changing dynamical conditions that barotropic perturbations encounter during the year and serves as a standard against which to compare perturbation behavior when a time-varying basic state is present. In section 4, we analyze the structures and time evolutions of some of the fastest-growing FTNMs by comparing them with the properties of normal modes with stationary basic states. In section 5, we attempt to understand the results of section 4 in a simpler context; we present a study of the instability of basic states that are scaled versions of the annual mean basic state with time-dependent scaling coefficients. In section 6, we consider the sensitivity of our results to changes in the basic states and to the strength and form of the dissipation. Our conclusions are summarized in section 7. In appendix A we formulate spectral representations of modes and in appendix B we present a semianalytical Floquet theory in which the space and time dependencies of the stability matrix are separable.
2. Theory and model details
In this section we present details of the barotropic model used in this study and summarize essential details of finite-time normal modes for the case when the basic states are periodic and hence allow the application of Floquet theory (Yakubovich and Starzhinskii 1975).
a. Linear barotropic model
To consider the instability of time-dependent basic states constructed from observations, we follow Simmons et al. (1983) and focus exclusively on barotropic instability within the framework of the nondivergent barotropic vorticity equation. By doing so, we effectively filter out the generally more rapidly growing baroclinic instabilities (Frederiksen 1983) and neglect topographic effects in the perturbation equation, which have been found to be small (Frederiksen and Bell 1990).
The linearized barotropic vorticity equation may be written as
Here ψ(λ, μ, t) is the perturbation streamfunction, ψ(λ, μ, t) is the basic-state streamfunction, ∇2 is the Laplacian operator, D is a dissipation operator, λ is longitude, μ is sine (latitude), t is time, a is the earth’s radius, and Ω is the earth’s angular velocity.
As in earlier studies of barotropic instabilities, we represent the basic state and perturbation fields using a spherical harmonic basis truncated rhomboidally at wavenumber 15 (appendix A). Our dissipation D is a −η∇2 operator applied to the vorticity, ∇2ψ, but acting only on spectral components whose total wavenumber n is greater than 15. Throughout this paper (apart from in section 6) we use a viscosity coefficient η = 2.5 × 105 m2 s−1. This scale-selective outer rhomboid dissipation, with this choice of eddy viscosity, is the standard formulation of diffusion in a number of general circulation climate and numerical weather prediction models (e.g., Bourke et al. 1977; Hart et al. 1990; Frederiksen et al. 1996). Rayleigh damping can also be accounted for through a term, κ∇2ψ, where κ is the drag coefficient. In fact, including the drag term just reduces the growth rates of normal modes (Simmons et al. 1983) and FTNMs by κ and has no effect on the modal structure or frequency. For this reason, our results will be presented for κ = 0 except in section 6 where we shall discuss the sensitivity of our results to the strength and form of the dissipation.
We now let x and x denote the column vectors of spectral coefficients for the perturbation and basic state streamfunction, respectively. Then the tangent linear spectral equations can be written in the form
Here, the matrix operator M(t) = M[x(t)] depends on the time-evolving basic-state flow, and we suppose that at the initial time, t = 0, x(0) is specified. The solution to Eq. (2.2) may be written as
where G(t, 0) is the propagator. Here, I is the unit matrix and T is the chronological time-ordering operator that orders the (noncommuting) matrices M(si) such that s1 ≥ s2 ≥ · · · ≥ sρ. The method of generating the matrix form of the propagator is described in Frederiksen (1998).
To discretize (2.4), we use a predictor–corrector time stepping scheme with a time step of ½ h. Basic states are formed by linearly interpolating in time between monthly climatological states, each of which is taken to be valid on the 15th day of the month. Except in section 6, where other periods are considered, the monthly mean states are based on 17-yr averages (1 July 1976–30 June 1993) of twice daily National Centers for Environmental Prediction–National Center for Atmospheric Research reanalysis 300-mb streamfunction fields.
b. Finite-time normal modes
The natural generalization of normal modes to the case of time-dependent flows are the eigenvectors of the propagator. We follow Frederiksen (1997) and call the propagator eigenvectors FTNMs to distinguish them from the modes of time-independent basic states, which we refer to as normal modes. Thus, for the propagator between the initial time t = 0 and a final time T, we have the eigenvalue–eigenvector problem
where λυ = λυr + iλυt are the eigenvalues, ϕυ the column eigenvectors of spectral coefficients, and N is the total length of the vectors. We can also define the growth rate ωυt and the phase frequency ωυr through the relationship
by analogy with the case of normal modes when M is time independent. Thus,
where arctan is a multivalued function; its appropriate branch needs to be chosen (Strong et al. 1995). The global growth rate ωυi gives the average rate at which the corresponding FTNM grows during the time interval T. Local or instantaneous growth rates of FTNMs can be functions of time as discussed in sections 3c and 4.
Frederiksen (1997) notes that there are three generic types of FTNMs. FTNMs with λυi ≠ 0 occur in complex conjugate pairs and are the natural generalization of traveling normal modes. FTNMs with λυi = 0 occur singly. When λυi = 0 and λυr > 0 the FTNM is “recurring,” because its structure at t = 0 repeats at t = T. When λυi = 0 and λυr < 0 the FTNM is a “flip” FTNM since its sign reverses at t = T though its structure is the same as at t = 0. All three types of generic FTNMs change structure between t = 0 and t = T. In this sense all three generic types of FTNMs appear to be traveling disturbances.
If the basic state, and hence the stability matrix M(t) = M[x(t)], is periodic then the linear system satisfies the conditions for Floquet theory (Yakubovich and Starzhinskii 1975; Seydel 1988; Iooss and Joseph 1990; Grimshaw 1990; Strong et al. 1995). Thus, if
where ρ is a positive integer. This means that as ρ → ∞, all initial disturbances (except for a set of measure zero) converge to G(t, 0)ϕ1[T, 0], the fastest-growing FTNM. The FTNMs play the same roles in the periodic system as do the normal modes in a linear system with a time-independent basic state or the Lyapunov vectors for a nonperiodic chaotic system (Szunyogh et al. 1997;Frederiksen 2000). Studying the leading FTNMs is a way of investigating the dynamics of those structures that, all else being equal, will be prevalent. For this reason we will always order modes by their growth rates and υ will reflect this ordering.
3. Normal modes for stationary basic states
In this section we examine the normal-modal instability of basic states consisting of (time invariant) climatological mean 300-mb streamfunction fields for each month of the year. Documentation of the seasonal dependence of normal-mode instability is of interest as an indication of the effect of seasonality on disturbance growth. Furthermore, when compared to the FTNM calculations of the next section, it will make it possible to determine whether instabilities available to the system are fundamentally altered when the continuous seasonal evolution of the climate state is taken into account. We note that the FTNMs described in section 2 reduce to normal modes when the basic states are time independent (Frederiksen 1997).
a. Growth rates and periods
The global growth rates ωi and the phase frequencies ωr as well as the corresponding e-folding times τi and periods Tr, for some of the fast-growing normal modes, are shown in Figs. 1a and 1b, for each of the stationary basic states for January through December. Also shown are the corresponding values for the annual mean basic state obtained by averaging the monthly average basic states. Among the fast-growing modes we have selected modes with periods greater than 15 days for display in Fig. 1. Included are the two fastest-growing modes in each month except for mode 1 for August (which has a period of 5.4 days and is therefore not shown in Fig.1b) as well as a selection of modes of interest in succeeding sections. With the exception of mode 1 for August, the structure of the corresponding eigenfunctions have significant amplitudes in the Northern Hemisphere. We note from Fig. 1 that though growth rates are not a smooth function of month, there is a tendency for relatively weak growth in northern summer and early fall, stronger growth in winter and spring, and strongest growth in late fall. These tendencies will be prevalent throughout the study.
Among the modes shown in Fig. 1 there are dominant modes with periods in the intraseasonal range, between 25 and 60 days, as well as longer-period modes including stationary modes with Tr = ∞. There are also, mainly subdominant, modes with periods shorter than 25 days, but for our analysis of FTNMs these will not be important.
b. Structures of normal mode disturbances
Next, we consider the structure of some of the normal modes whose growth rates and phase frequencies are shown in Fig. 1. We focus on modes that are of particular interest for comparison with FTNMs considered in section 4 and that are representative of the effects of seasonality on modal structure. Figure 2 shows the 300-mb disturbance streamfunction, at a particular phase, denoted phase 0 (and in some months at a phase of 90°), for the fastest-growing modes for January, March, July, and December monthly averaged basic states and as well for the annual mean basic state. Also shown is mode 5 for the April monthly averaged basic state, which, as shown in section 4, has largest pattern correlation with FTNM1 in April. For April, July, and December the modes shown in Fig. 2 have periods in the traditional range between 30 and 60 days associated with intraseasonal oscillations while the other displayed modes have periods longer than 100 days.
The winter modes shown in Fig. 2 are reminiscent of prominent teleconnection patterns in that they have large scale, cover much of the Northern Hemisphere, and often have components resembling wave trains. They share these properties with the leading Northern Hemisphere winter barotropic modes studied by Simmons et al. (1983) and with the large-scale barotropic and equivalent barotropic modes examined by Frederiksen (1983) in barotropic and baroclinic models. Modes in other seasons are distinct from these in terms of their scale and the prominence of Southern Hemisphere features but not necessarily in terms of growth rates. For example, the July mode has a dominant zonal wavenumber 3 structure in the Southern Hemisphere (cf. Mo and White 1985) and small-scale Northern Hemisphere features clustered near the pole.
c. Local growth and intramodal interference effects
The global growth rate, ωυi, only gives the average growth of a mode over a full period Tυr or averaged over all its phases. The complicated structural changes of the normal-mode disturbances seen in Figs. 2a,b and Figs. 2c,d, may result in variations in disturbance growth at a given geographical location (Frederiksen 1979; Simmons et al. 1983) and for disturbance norms (Penland and Sardeshmukh 1995; Borges and Sardeshmukh 1995) during a mode’s evolution. In this section we show that for some normal modes the transient bursts of growth and decay in the L2 streamfunction norm, relative to the global growth rate, may be quite large. When comparing normal-mode growth to FTNM growth in the next section, these bursts must be taken into account so here we quantify their magnitude.
represent the time-dependent column vector of the υth eigenfunction spectral coefficients [Eq. (A.2)] and define the streamfunction L2 norm of x by
where + denotes Hermitian conjugate and T denotes transpose. Thus, from Eq. (A.2),
where αυ(t) = ωυrt and ϕυmn is the time-independent spectral coefficient of the υth eigenvector with zonal wavenumber m and total wavenumber n.
The L2 norm squared thus consists of a nonoscillatory contribution that grows exponentially with growth rate 2ωυi and an oscillatory component that oscillates with time as it grows in magnitude (cf. Frederiksen 1979). The oscillatory component manifests itself in Eq. (3.3) through intramodal interference effects between the eastward (e.g., ϕυmn) and westward (e.g., ϕυ−mn) propagating components of the mode. These interference effects only occur for traveling modes for which ωυr ≠ 0.
To quantify the time dependence of modal growth we use two measures, the local total growth rate, ω̃υi(t), and the relative amplification factor, Rυ(t). The total growth rate is defined from the total amplification factor Aυ(t) = ‖xυ(t)‖/‖xυ(0)‖ by
Here the local relative growth rate, ω̂υi(t), gives the departures from the global rate; it is related to Rυ(t), the relative amplification factor, through
Alternatively, Rυ(t) can be viewed as an integral measure of ω̂υi(t).
As an example of a mode with large fluctuations in local growth, Fig. 3 displays ω̃1i(t) and R1(t) for mode 1 for the March basic state as functions of time and phase. Both measures vary by more than a factor of 4 as the mode evolves. Though the range of local growth rates attained by this mode is especially large, the values of FR = maxt(Rυ(t))/mint(Rυ(t)), mint(ω̃υi(t)), ωυi, and maxt(ω̃υi(t)) shown in Table 1 indicate that significant variations are common. Table 1 presents results for modes 1 and for other modes with high pattern correlations with FTNM1 of section 4 (Fig. 6).
4. FTNMs for time-dependent basic states
a. Time-dependent basic states
Having established the significant effects of seasonality on modes within a single month we now determine the effect of the progression of climatological states on modal characteristics. We do this by considering FTNMs for the time-dependent basic state described in section 2a, which represents the annual cycle and has a recurrence time T of 360 days.
Before presenting the FTNMs for the annual cycle of basic states, it is worthwhile to consider what kind of behavior we might expect them to display. We know that, for an unstable linear system with a time-independent basic state, almost all initial disturbances will converge to the fastest-growing mode after sufficient time. Consider now a time-dependent basic state that is changing very slowly. Then we expect FTNM1, evolving in the tangent linear model, to take up the structure of mode 1, appropriate to the local basic state. If mode 1 is a traveling mode then FTNM1 will also undergo structural changes and variations in its local total growth rate. On the other hand, for more rapid changes in the basic state, we expect departures from this limiting case. Departures will occur because when the evolving mode first experiences the newly changed basic state it will already have been influenced by the previous basic states (an effect we will call preconditioning) and thus will not have the structure of the fastest-growing mode for the new basic state. In this case, FTNMs will be composed of a superposition of normal modes, but because of differential growth rates, evolution will be biased toward the fast-growing modes of the changed basic state. Of special interest is the possibility that preconditioning can lead to FTNM growth rates that are larger than the global or even local growth rates of the associated leading normal modes in a similar way to transient supermodal growth of singular vectors (Farrel 1989; Frederiksen 2000 and references therein).
b. Growth rates and periods
Table 2 shows the eigenvalues λυ = λυr + iλυi and mode type for the five fastest-growing FTNMs as well as their corresponding global growth rates, e-folding times, and periods. The amplification factors |λυ| give the average amplification of the FTNMs over a one-year period; they are reduced by drag by a factor exp(−κT). Since a given disturbance is only likely to approach FTNM structure for part of the seasonal cycle, the amplification factors merely provide an indication of which structures are dynamically favored, through the associated growth rates. As mentioned in section 2b, the amplifications of the FTNMs are not uniform but vary with time. We define the relative amplification factor Rυ(t) for an FTNM associated with eigenvector ϕυ as follows. At t = 0, the streamfunction spectral coefficient for this FTNM again has the form in Eq. (A.2) and xυ(0) is the column vector of spectral coefficients as given in Eq. (3.1). Then Rυ(t) is again given by Eq. (3.6) with
Figure 4a shows the local total growth rate, ω̃1i(t) (dashed) and R1(t) (solid) for FTNM1. Also shown is ω1i (dotted) for normal mode 1 calculated daily using the interpolated basic state of section 2a on each day. This latter curve includes on the 15th of each month the global growth-rate information contained in Fig. 1a for mode 1 for each month. In this figure and others concerning the annual cycle, we consider the monthly averages from which the basic state is formed to represent the 15th of the month. We have chosen the initial time t = 0 in the calculation of FTNMs to be 15 January. However, the same FTNMs would be obtained on choosing any other initial time.
The local total growth rate of FTNM1 generally follows the global growth rate of normal mode 1, with large values in late boreal fall and early winter and small values in summer and early fall. As would be expected because of preconditioning and intramodal effects, at times there are marked differences between the global normal-mode growth rates and total FTNM growth rates. For example, FTNM1 is growing much more rapidly than normal mode growth rates in March and December while it is much slower in April and October/November. In section 4d, we examine the causes of these differences.
The relative amplification factor of FTNM1, which is an integrated measure of the relative growth rate, shows a large increase from November through December–January. There is a further large increase in March prior to the maximum in R1(t) in early April, followed by a decrease throughout the rest of the year.
The characteristics displayed in Fig. 4a turn out to be representative not just of FTNM1 but for the class of leading FTNM perturbations under the influence of the annual cycle. Figure 4b shows the average local total growth rate of the five leading FTNMs (in dashed) and the relative amplification factor R(t) (in solid) corresponding to the average relative growth rate. Also shown is the average global growth rate of the five dominant normal modes (in dotted) calculated daily using the interpolated basic state of section 2a on each day. Again, the maximum local total growth rates occur in November and December with secondary maxima in late winter and spring and a minimum in October. This seasonality is also found for averages over the 10 leading FTNMs (not shown).
We expect that the dominant FTNMs will have a dynamical advantage in the selection and structural organization of large-scale barotropic disturbances (Frederiksen 2000). However, a given disturbance is likely to approach FTNM structure for only part of the seasonal cycle. Disturbances may have relative growth rates with seasonal variations similar to those in Figs. 4a and 4b; we note that relative growth rates are independent of the drag coefficient κ of section 2. However, we expect that the relative amplification1 of a given disturbance will only follow the general curves in Figs 4a and 4b during parts of its evolution.
The annual average growth rate of FTNM1 is 0.059 day−1 (Table 2) which is 9% less than the annual average of the daily global growth rates of mode 1, which is 0.065 day−1. Comparison of normal mode and FTNM growth rates at instants of time is complicated by intramodal growth effects, but the systematic reduction in the growth rate of FTNM1 from the average growth rate of normal mode 1 is almost certainly related to the preconditioning of the modal structure required for efficient growth of the FTNM for time-dependent basic states. That is, what is important is not just the growth rate of the dominant normal mode during a given month (assuming a time-independent basic state), but also whether the modal structure growing fastest during the previous month is compatible with the modal structure during the month of interest.
By contrast, the average growth rate of the leading five FTNMs (Table 2) and the annual average global growth rate of the leading five normal modes for the daily basic states (from dotted line in Fig. 4b) both have a value of 0.044 day−1. Thus, the leading FTNMs are able to adjust to the changing basic state in such a way that on average they maintain the maximum global growth available.
c. Structure and evolution of FTNM disturbances
In this section we discuss the structures of the fast-growing FTNMs focusing on FTNM1 and its evolution when integrated forward in the tangent linear equation (2.2). As was the case in our analysis of the growth of FTNM1, we are especially interested in determining the degree to which the evolving FTNM structure can be understood in terms of the structure of the normal modes of section 3.
Figure 5 shows the 300-mb streamfunction for FTNM1 at various phases through the year. The results in this figure are divided by the growth factor exp(ω1it). We see that the structures vary considerably from month to month and we have found that they often resemble one phase of normal modes in corresponding months. To quantify this similarity we have calculated pattern correlations between FTNM1 on the 15th of each month and each of the 10 leading corresponding normal modes of the monthly mean states. This correlation depends on the normal mode phase. Figure 6 displays the pattern correlation and normal-mode index when the correlation is maximized over all phases and all 10 modes. In all months there are normal modes for the monthly averaged basic states that have relatively large pattern correlations with FTNM1 for the time-varying basic state. For example, on 15 January FTNM1 has a correlation of 0.91, which occurs for phase 0 of the January leading normal mode (Fig. 2a).
In January, February, March, June, July, November, and December, FTNM1 has largest pattern correlation with mode 1, but for the other months subdominant modes yield the biggest correlation. Thus, it seems that it is not always possible for FTNM1 to change its structure sufficiently quickly between the months to appear most like mode 1. As explained above, one might suppose that the selection of mode 1 structure by FTNM1 would be most likely when the structure of mode 1 changes least between consecutive months; for larger changes in mode 1, the selection of subdominant modal structures by FTNM1 may occur. To examine this proposition, we have also presented in Fig. 6 pattern correlations, maximized over all phases, between mode 1 in a given month with mode 1 in the following month (dot-dashed) and between mode 1 in a given month with FTNM1 in the following month (dashed). Figure 6 shows that the largest correlation is between modes 1 for December and January and the second largest for November and December, consistent with FTNM1 having largest correlation with mode 1 during these three months. For smaller lagged pattern correlations between modes 1 we see from Fig. 6 that it may be possible for FTNM1 to select a subdominant modal structure. We also note that the high lagged correlations between mode 1 and FTNM1 in November, December, and January indicate the normal mode 1 in these months would be a suitable precursor for FTNM1 in the following month.
To make the time evolution of FTNM1 (scaled by expω1it) more evident we show in Fig. 7a a longitude–time cross section along 60°N. This figure makes clear that though FTNM1 repeats its structure only every two years, its intraseasonal variability is significant. A power spectrum analysis of the time series along 60°W (not shown), for example, indicates that the peak of the spectrum occurs at a period of 150 days with a secondary maximum just below 60 days. These periods are another reflection of the influence of the dominant monthly normal modes on FTNM1. The presence of intraseasonal variability in our results, with an observation-based basic state, reinforces those of Strong et al. (1995), who found intraseasonal winter oscillations (with slightly shorter period than ours) in an idealized time-varying basic state. The local period of FTNM1 in winter is also only slightly longer than the 40-day period of the fastest growing mode in the study of Simmons et al. (1983).
Figure 7b shows a latitude–time cross section of the zonal average of the absolute value of the 300-mb streamfunction for FTNM1 (scaled by expω1it). Figures 7a and 7b reflect the seasonal variation of the relative amplification factor in Fig. 4a. Figure 7b also shows that the largest equatorward penetration of FTNM1 occurs in boreal spring when the relative amplification factor is also the largest.
As was true for growth rates, the structure of FTNM1 is also representative of the general structure of the class of dominant FTNMs. All FTNMs of Table 2 are large-scale disturbances that exhibit seasonal changes in their structures. For example, FTNM2 has pattern correlations with FTNM1 that range between 0.8 and 0.95 between August and January, drop to 0.7 in February, and then to lower values during the remaining months.
d. Local growth and intramodal and intermodal interference effects
Next, we examine the roles of intramodal growth and intramodal interference effects, as well as intermodal interference effects, in the growth of FTNM1, focusing on the development in boreal spring.
Table 3 lists the largest pattern correlation between FTNM1 and mode 1 for March (15) at 3-day intervals between 6 March and 15 April. Also shown is the phase of mode 1 for March for which the pattern correlation is a maximum. We note that the phase increases monotonically from 0° on 6 March to 170° on 15 April, with a phase of 90° occurring near 3 April. Throughout this period the pattern correlations are close to or greater than 0.6 with values reaching 0.94 near the middle of the period. This suggests that to a first approximation, the behavior of FTNM1 during boreal spring is essentially that of mode 1 for March during its first half-cycle (Fig. 3) with bursts of growth and decay related to intramodal behavior.
To determine whether intermodal interference is also contributing to the extreme growth of FTNM1 near 20 March, we have found the maximum local growth rate for every 20 March mode throughout its evolution. It turns out that even the largest of these is about 15% less than FTNM1’s 20 March growth rate and intermodal interference is necessarily at work. This verifies that intramodal effects by themselves are not sufficient to explain the growth of FTNM1. Similar comparisons indicate that there are several other times during the annual cycle when intermodal interference effects are the only way that FTNM1 could attain the local growth it experiences.
To further verify the role of intermodal interference in mid-March FTNM1 growth, we have expanded the unit norm FTNM1 on 15 March in terms of the unit norm normal mode eigenvectors for the 15 March basic state and found that the absolute value of the projection coefficient for the projection of FTNM1 onto eigenvector 1 is 1.53. This indicates the significant role of intermodal interference effects with the subdominant modes destructively interfering with mode 1 to reduce the norm of FTNM1 from 1.53 to unity.
By contrast, the changing basic state slows growth soon after mid-March. If the structure of FTNM1 on 15 March is integrated forward under the influence of a fixed 15 March basic state, its growth remains above its starting value of 0.99 day−1 until 29 March, while Fig. 4a shows the FTNM1 growth rate plummets with the annual cycle basic state. In a similar way, we have shown that the reduction in growth rate of FTNM1 during the second half of December (Fig. 4a) is due to the changing basic state being less favorable to growth than the 15 December basic state.
5. FTNMs for time-dependent scaled annual mean basic states
a. Time-dependent scaled annual mean basic states
In the previous section, we found it useful to interpret FTNM1 behavior by relating it to normal modes of time-independent basic states. In this section we shall attempt to understand FTNM1 behavior by comparing it to modes of simplified systems with time-dependent basic states. Here, we first reduce the complexity by simplifying the basic state. We test the hypothesis that, for instabilities, the most important aspect of the seasonal cycle is likely to be the amplitude of the mean state, by considering a basic state that has a structure that is fixed in time though its amplitude has temporal variations. This basic state is produced exactly as the basic state in section 4 was generated, except rather than basing it on monthly means, we represent the midpoint of each month by a constant times the annual mean 300-mb streamfunction. That constant is found by performing a least squares fit of the scaled annual mean state to the observed monthly mean state in the Northern Hemisphere. This means that the basic state for each month is replaced by a scaled version of the annual mean basic state. The scaling coefficients cj for each month are given in Table 1. We have chosen to do the least squares fits only over the Northern Hemisphere since our instabilities are primarily Northern Hemisphere instabilities. However, we use the global scaled annual mean basic states in all our calculations. All other aspects of the modal calculation are as in section 4.
b. Growth rates and periods
The fastest-growing FTNM for the time dependent scaled annual mean basic state is a “flip” mode with τi = 27.0 days and Tr = 2 yr. FTNMs 2 and 3 are recurring modes with e-folding times of 29.3 and 31.0 days, FTNM4 is a flip mode with τi = 32.4 days and FTNM5 is a generalized traveling FTNM with τi = 32.5 days. Comparing these results with those in Table 2, we notice that the e-folding times are longer with the time-dependent scaled annual mean basic state than with the time-dependent basic state obtained from the monthly averaged basic states. This presumably reflects the smoother potential vorticity gradients associated with the extra averaging involved in forming the annual mean basic state.
Figure 8 shows the local total growth rate, ω̃1i(t) (dashed), and the relative amplification factor R1(t) (solid) for FTNM1 for the time-dependent scaled annual mean basic state. As expected, prior to each local maximum in the relative amplification factor there is a period of relatively large local total growth rate. The bursts of growth are to be expected on the basis of the discussion in section 3c and the variations in the local total growth rate for mode 1 for the annual mean basic state (Table 1). The maximum relative amplification factor now occurs just before 15 May, with a secondary maximum around 15 March. This compares with the maximum in R1(t) occurring around the first of April for the time-dependent basic state obtained from the monthly averaged basic states (Fig. 4a). In both cases, the maximum relative amplification factors occur in boreal spring with minima in boreal fall. The other four fast-growing FTNMs discussed above also have relative amplification factors that have maxima in the first half of the year and minima in boreal fall (cf. Fig. 4b).
The average growth rate of FTNM1 in this case is 0.0370 day−1, 6.5% less than the growth rate for mode 1 for the annual mean basic state, which is 0.0396 day−1. Again the required changes in modal structure of FTNM1 results in a reduction in the average growth rate.
We shall not detail the structural changes of the FTNM1 disturbance streamfunction other than to note that we have constructed a longitude–time cross section of this field at 60°N as well as latitude–time cross sections of its zonally averaged absolute value. When compared to the corresponding displays in Fig. 7, we find that there are features, including the Northern Hemisphere shift from monthly timescale oscillations in winter and spring to nearly standing variability in summer and fall, the large equatorward penetration in boreal spring, and the overall seasonality in amplitude, which are much the same with the simplified basic state as they were with the complex basic state of section 4.
c. Analytical results
We show in appendix B that it is possible to obtain a semianalytical solution for the FTNMs by simplifying the propagator still further. We allow the diffusion and beta effect to also be scaled by the cj in Table 1. Then the FTNMs have the horizontal structures of the corresponding normal modes of the annual mean basic state but with amplitudes that vary with time. As discussed in appendix B, traveling FTNMs have local growth rates that depend on phase. Interestingly, if phase effects are removed by averaging over all phases then the local growth rates are simply proportional to the time-varying amplitude of the basic state [Eq. (B.3)]. The dotted line in Fig. 8 shows the relative amplification factor of FTNM1 with phase averaging [R1NO(t) of Eq. (B.2)]. Because FTNM1 is a traveling eigenmode it makes excursions about R1NO(t) associated with intramodal interference effects that depend on the initial phase of the eigenmode, as discussed in appendix B. It is clear that this highly simplified case captures the essence of the more complex models in sections 4 and 5 and demonstrates the link between the seasonality of growth and the annual cycle of mean circulation amplitude.
6. Sensitivity studies
The structure and growth rates of boreal winter normal-mode disturbances have been found to be sensitive to the basic state and dissipation used in modal calculations (Simmons et al. 1983; Anderson 1991; Frederiksen and Frederiksen 1993; Borges and Sardeshmukh 1995). To verify that our conclusions concerning seasonality are valid in spite of this sensitivity, we have tested their robustness to reasonable changes in these parameters. Here we summarize these results by focusing on the sensitivity of seasonal variations of growth rates of normal modes and FTNMs.
As an interesting test of sensitivity to basic-state specification, we have found how our results are affected by the decadal shift in the observed mean state that occurred in the mid-1970s (Trenberth 1990). Figure 9a shows the average global growth rates, taken over the five fastest growing normal modes, for basic states based on climate average 300-mb streamfunction fields as described in section 2a, but now also considering other time periods in addition to the standard 17 yr from 1 July 1976 to 30 June 1993. The other two cases are for the 20 yr beginning in 1958 and the 20 yr beginning in 1978. The growth rates for individual months can differ by as much as 25% between the basic states, but the distinct seasonal cycle found earlier is evident in each of the three datasets. Growth rates are largest in November, December, and January; they decrease in northern summer and are generally small in October.
For comparison, we shown in Fig. 9b 30-day running means of the average local total growth rates, taken over the five fastest-growing FTNMs, for the corresponding time-dependent basic states based on the 17-yr and two 20-yr datasets (linearly interpolated as described in section 2a). For the 17-yr standard dataset, Fig. 4b shows the corresponding growth rates without the 30-day time averaging. For each of the datasets, the largest FTNM growth rates also occur in late northern fall with winter again giving larger growth rates than summer. In these respects, there are close similarities between the average normal mode and average FTNM growth rates. This seasonal variation in growth rates of normal modes and FTNMs has been found to be robust for other long-time averaged datasets of 300-mb streamfunction fields and for averages taken over the 10 leading modes.
Our tests show that the seasonal cycle of modal behavior is also insensitive to dissipation specification. As noted in section 2a, the only effect of including the drag term is to reduce the growth rates of normal modes and FTNMs by the drag coefficient κ, but provided the drag is not a function of season this will not affect the seasonality we found. Similarly, halving or doubling the scale-selective dissipation rate η in our model, or changing the dissipation to act on the full rhomboid, changes none of the general conclusions concerning the effect of seasonality on the growth of perturbations.
7. Discussion and conclusions
The purpose of this article has been to examine the effects of seasonality of the mean atmospheric flow on the growth and structures of large-scale barotropic disturbances. We have focused on two limiting cases in which general disturbances can be naturally represented as superpositions of eigenmodes. In the first, the perturbations see the mean basic state in a given month but do not experience the month to month changes in the basic flow. This leads to the usual normal modes that have been widely studied for northern winter mean flows as discussed in the introduction. In the second case, the disturbances encounter the full annual cycle of variations in the basic state, which is taken to be periodic with a period of one year. This allows the application of Floquet theory and results in basis functions which we call finite-time normal modes (FTNMs). Although details of the properties of the class of dominant normal modes and FTNMs may differ, their variations in growth and structure follow similar seasonal cycles. In particular, average growth rates of the 5–10 dominant normal modes and of the corresponding FTNMs are largest in November, December, and January, decrease in boreal summer and are generally small in early and mid-fall. We can therefore be fairly confident that general large-scale barotropic disturbances that growth for a finite time under the influence of the seasonally varying basic state will also experience this same seasonal variability in growth.
All of the dominant FTNMs that we have examined are large-scale modes with generally largest amplitudes in the Northern Hemisphere. They exhibit intraseasonal variability as well as longer period variations. In boreal winter, they have similar scales to the dominant barotropic and equivalent barotropic normal modes found in the instability calculations of Simmons et al. (1983) and Frederiksen (1983). In all months, there are also dominant normal modes of our monthly averaged stationary basic states that have relatively large pattern correlations with FTNM1 for the time-dependent basic state. These pattern correlations are highest in December and January and lowest in April and October. We have examined the extent to which changes in eigenmode structure affect the growth rate of FTNM1. We find that the average growth rate of FTNM1 is reduced by 9% compared with the annual average of the daily global growth rates of the fastest-growing normal modes. By contrast, the average growth rate of the five leading FTNMs is practically the same as the annual average global growth rate of the five leading normal modes for the daily basic states. That is, the leading FTNMs are able to adjust to the changing basic state such that on average they achieve the maximum global growth available.
In view of the strong correlations between FTNM1 and modes of the monthly mean states, we have found it useful to understand FTNM behavior in terms of these modes. We have examined the roles of intramodal growth and intramodal and intermodal interference effects in the evolution of FTNM1 focusing on the development of FTNM1 during boreal spring. We have found that traveling normal modes of the monthly basic states exhibit significant transient bursts of growth and decay (of their norms) relative to their global growth rates. We find that they are particularly significant for the March basic state where the relative amplification factor changes by a factor of 4 during the evolution of mode 1. Local bursts of growth and decay carry over to the behavior of leading FTNMs. For example, we have examined in detail the dramatic relative amplification and decay of FTNM1 in boreal spring and found that, to a first approximation, its evolution is essentially that of mode 1 for March (15) during its first half-cycle of relative growth and decay involving intramodal interference effects. However, we have established that intermodal interference effects also play an important role resulting in supermodal growth around 20 March. After mid-March the rapid relative decay of FTNM1 is found to be due to the changing basic state being less favorable to growth.
We have also tried to understand the behavior of the FTNM instabilities within simplified models. First, we have reduced the complexity of the basic state by considering a mean flow that has the horizontal structure of the observed annual mean basic state but scaled by a time-dependent scaling factor that varies with the annual cycle. Second, we have developed an analytical Floquet theory of periodic basic states in which space and time dependencies are separable by also allowing the dissipation and beta effect to be scaled by the same time-dependent scaling factor. Both simplified FTNM calculations indicate that much of the nonuniform growth found in the complete FTNM analysis of the annual cycle is explicable by seasonal variations in the amplitude of the mean state.
We have examined the sensitivity of our results to reasonable changes in the basic state and strength and form of the dissipation. We have focused on the sensitivity of seasonal variations of average growth rates of normal modes and FTNMs taken over the 5–10 dominant modes. The growth rates in individual months, for different long-time averaged datasets of 300-mb streamfunction fields, can differ by as much as 25% between these datasets; but the same distinct seasonal cycle is evident for all the datasets. The strength of the dissipation rate and drag term will determine the growth rates of perturbations in a given month. However, we find that the seasonal cycle of relative growth of modes is also insensitive of the damping specifications we have considered.
Modal analysis may be relevant to both initial value and forced problems (Branstator 1985; Frederiksen and Webster 1988). Indeed, there is a general consistency between our results and those of Newman and Sardeshmukh (1998), who consider the linear response to forcing in the barotropic vorticity equation. Their regionally integrated root-mean-square height anomalies and lagged pattern correlations (their Figs. 7 and 8) show a generally similar seasonal variation to the increases and decreases in magnitude of our relative amplification factors (Figs. 4a,b).
A controversial issue in atmospheric dynamics has been whether disturbances primarily develop as exponentially amplifying normal modes or whether transient development involving intermodal interference effects is generally important (Farrell 1989; Frederiksen and Bell 1990; Whitaker and Barcilon 1992; Borges and Sardeshmukh 1995; Frederiksen 2000 reviews the literature). Singular vectors (SVs), also termed optimal perturbations, have been proposed as prototypes that have a role in the theory of transient development similar to that of normal modes in exponential instability theory (Farrell 1989). For time-independent basic states the leading adjoint mode is equivalent to the leading singular vector optimized for a long time. Frederiksen and Bell (1990), for example, considered the roles of adjoint modes and normal modes in the instability and error growth during North Atlantic blocking in January 1979. They found fast-growing large-scale equivalent barotropic modes located in the blocking region over the North Atlantic on 20, 21, and 22 January. As well, they noted that the leading adjoint mode in the streamfunction norm on 20 January was a small-scale disturbance reminiscent of a localized storm off the east coast of North America. As a consequence they hypothesized that the intense and localized cyclogenesis that occurred around 20 January would preferentially excite a large-scale anomaly in the North Atlantic blocking region. This sequence of events leading to large-scale anomalies is also very similar to that found on the basis of an observational study by Colucci (1985) and in a subsequent instability analysis by de Pondeca et al. (1998).
The adjoint modes and SVs are, however, not unique but depend on the chosen norm. It therefore seems to be of considerable interest that we have found that interference effects may also play a very significant role in the evolution of norm-independent structures such as normal modes and FTNMs. These results add support for the case that rapid transient development due to interference effects may be a common feature of dynamical systems such as the atmosphere.
Finally, we consider the possibility that the seasonal cycle of modal growth found in this study may be related to the seasonal dependence of predictability found in some models of climate prediction over the tropical Pacific Ocean (Latif and Graham 1991; Cane 1991). These models encounter a predictability barrier in boreal spring when correlations between observations and predictions rapidly decline. Lagged correlations between the mean monthly Southern Oscillation index are also found to decrease rapidly in boreal spring (Webster and Yang 1992; Webster 1995). We have found that relative amplification factors of large-scale dominant FTNM instabilities increase significantly in the late northern winter and early spring, followed by a decrease in late spring and early summer. The boreal spring peak is also associated with the maximum equatorward penetration of these disturbances. In a future study we hope to examine whether a contributing cause of the boreal spring predictability barrier may be the fact that the amplitudes of large-scale instabilities of the atmospheric circulation have peaks in boreal spring.
It is a pleasure to thank Tony Davies, Steve Kepert, and Andy Mai for assistance with this work. One of us (JSF) wishes to thank Joseph Tribbia for organizing a very enjoyable extended stay with the Climate and Global Dynamics Division, National Center for Atmospheric Research, while the other of us (GB) expresses his thanks to the staff of the Cooperative Research Centre for Southern Hemisphere Meteorology and CSIRO Atmospheric Research for their contribution to his productive visit to these institutes. Partial support for this project was provided by Contract NA76GP0366 with the Office of Global Programs of the National Oceanic and Atmospheric Administration.
We formulate our instability problems in spectral space. Each of the perturbation and basic-state streamfunctions is expanded in spherical harmonics; for example,
where J = 15, Pmn(μ) are orthonormalized Legendre functions, m is zonal wavenumber, and n is total wavenumber. If ϕυmn are the spectral coefficients of the υth eigenvector ϕυ, then the corresponding physical space streamfunction ψυ(λ, μ, t) has the form in Eq. (A.1) with
(Frederiksen and Bell 1990). Here, in general ϕυ−mn ≠ ϕυ*mn but ψυ−mn(t) = ψυ*mn(t), where * denotes complex conjugate, ensures that the physical space field is real. In Eq. (A.2) the phase angle αυ(t) is defined by
Analytical Floquet Theory
Here we derive an analytical Floquet theory of time-dependent periodic basic states in which the space and time dependencies are separable. The time-dependent basic state is again obtained from the scaled annual mean basic states representing the different months, as described in section 5a. However, in order to arrive at an analytically solvable model, we also need to allow the constant diffusion and beta effect to be scaled by the cj in Table 1 for the different months. It is in fact not necessary to scale the drag term (which we take to be zero here) because it commutes with the stability matrix. Since the cj only vary by about a factor of 2 over the annual cycle, we expect that the analytical model may capture the essence of the more complex models in sections 4 and 5.
Let us denote by c(t) the linear interpolation of the scaling factors cj between the different months. Then the stability matrix M(t) = c(t)Ma, where Ma is the stability matrix for the annual mean basic state. The propagator is again given by Eq. (2.4) but the time-ordering operator T is no longer needed since the matrices M(t) at different times now commute. The eigenvalue–eigenvector problem (2.5) is thus separable in space and time with the eigenvectors corresponding to the normal modes of the annual mean basic state. The norm squared of xυ(t) is again given by Eq. (3.3) but with the replacements
Traveling eigenmodes again produce bursts of local growth evident in the oscillatory component in Eq. (3.3). The analysis of stationary eigenmodes, and of the nonoscillatory contribution to the norm of travelling eigenmodes [Eq. (3.3)], is simpler and we consider these equivalent cases first. The nonoscillatory contribution to the relative amplification factor [Eq. (3.6)] of FTNM υ is
The associated growth rate is
For traveling eigenmodes, we need to add to gυ(t) the local relative growth rate of the FTNM; this depends on the initial phase of the eigenmode. A generalization of the argument of section 3 shows that if the maximum (minimum) local relative growth rate of a given FTNM occurs at a time t, then its value is c(t) times that of the maximum (minimum) local relative growth rate of the associated mode. For the annual mean basic state, the local relative growth rate of mode 1 varies between ±0.0191 day−1 (Table 1).
* Permanent affiliation: CSIRO Atmospheric Research, Aspendale, Victoria, Australia.
+ Permanent affiliation: National Center for Atmospheric Research, Boulder, Colorado.
Corresponding author address: Dr. Jorgen S. Frederiksen, CSIRO Atmospheric Research, Private Mail Bag No. 1, Aspendale, Victoria 3195, Australia.
Our relative amplification factor (3.6) is normalized to unity at t = 0 corresponding to 15 January. For disturbances initialized at other times of the year it may be more convenient to normalize the relative amplification factor to be unity at another time. In that case, the two relative amplification factors would differ by a scaling factor but the growth rates would be the same.
# The National Center for Atmospheric Research is sponsored by the National Science Foundation.