## 1. Introduction

The atmosphere, as a continuous medium, is governed by nonlinear partial differential equations based on the Navier–Stokes equations. These equations, when projected onto a specified basis, become equivalent to a dynamical system with an infinite number of degrees of freedom or, when a truncation is applied, to a finite dynamical system whose nonintegrable properties may produce chaos (Lorenz 1963; Lichtenberg and Lieberman 1983). This is the main feature of atmospheric predictability (Lorenz 1965), and it is now generally agreed that the atmosphere is not deterministically predictable beyond a few weeks. But the interaction between the different scales of the system and the energy redistribution (via energy cascade in 2D for example) produce a spectrum shifted toward the red indicating the existence of preferred “stagnation scales.” This fact has attracted increasing interest over the last decade or so because of the need to understand the predictable part of the atmosphere, such as low-frequency intraseasonal oscillations (Branstator 1987) related to large-scale persistent flow patterns. Such patterns have been observed in the midlatitude atmosphere and have been noticed to persist for time periods much longer than those typical of midlatitude cyclones (Namias 1950), that is, longer than the typical dominant timescales of baroclinic instability in the storm track regions but shorter than the intraseasonal variations caused by changes in radiative fluxes and surface boundary conditions such as SST anomalies. Many works have been devoted to identifying such persistent or quasi-stationary patterns or weather regimes, to analyzing the main sources of their maintenance by studying the corresponding high-frequency transient eddies, and to a lesser extent, to the study of their mutual transitions, onsets, and breaks. For example, Rex (1950) and Namias (1964) were the first to notice that blocking, considered as a regional persistent weather pattern (Dole and Gordon 1983; Dole 1986) over Europe can persist for several weeks.

A simple barotropic beta-channel model forced by bottom topography and thermal driving was used by Charney and Devore (1979) to show that multiple equilibria may exist. These multiple equilibria are represented by fixed points within the phase space and in their model showed two regimes: one with high (blocking) and the other low (zonal) index. The instability of these equilibria may explain intermittent persistent behavior, as pointed out by Legras and Ghil (1985), who defined quasi-stationarity through the balance in the tendency equation. States similar to the high and low index metastable equilibria of Charney and Devore (1979) and also Wiin-Nielsen (1979) were found later by Hansen (1986) and Hansen and Sutera (1986) in atmospheric data using a probabilistic approach. Their wave amplitude index used the combined rms amplitudes of zonal waves 2, 3, and 4 derived from the Fourier decomposition of a latitudinal band average of 500-mb geopotential height. They obtained two maxima in the probability density function (pdf) associated respectively with high- and low-wave amplitude regimes.

Although the previous findings of Charney and Devore (1979) and Hansen and Sutera (1986) were challenged by Tung and Rosenthall (1985) and Nitsche et al. (1994), the existence of multiple equilibria or weather regimes in the atmosphere seems to be likely in some form. The existence of free-mode Rossby waves may almost guarantee this. In fact, Wallace and Gutzler (1981) identified systematically a whole set of large-scale 500-mb teleconnection patterns based on 30 yr of National Center for Environmental Predictions (NCEP, formerly the National Meteorological Center) data using a simple correlation technique among which the Pacific–North America (PNA) and also the “east Atlantic” teleconnection patterns were found to be robust. This was shown to correspond well with results from other independent studies using rotated principal component analysis (Horel 1981; Barnston and Livezy 1987). Molteni et al. (1988) computed the pdf of their 500-hPa height EOF1 (identified as PNA) coefficients from a 32-yr observed time series and found a non-Gaussian behavior. In particular Molteni et al. (1990) showed that the pdf of the PNA index is bimodal. Previously, Palmer (1988) had also studied the skewness in the pdf of the PNA index. Many other studies based on recurrency and persistence properties have also shown the existence of weather regimes in the atmosphere (Dole 1986; Mo and Ghil 1988; Molteni et al. 1990). Vautard and Legras (1988) defined quasi-stationarity within a baroclinic quasigeostrophic channel, in a similar way to Charney and Devore (1979), by enforcing tendency balance at large scale only and identified two flow regimes, namely a zonal and a blocking regime. Vautard (1990) applied this method on 37 winters from NCEP and found four regimes over the Atlantic. The influence of the boundaries between these regimes on their mutual transitions has been studied by Hannachi and Legras (1995).

Various dynamical processes have been presented in an attempt to explain the existence and maintenance of these flow regimes; Rossby wave propagation from the Tropics as a response to thermal forcing (Hoskins and Karoly 1981), barotropic instability of a zonally asymmetric flow (Simmons et al. 1983), the existence of multiple stationary solutions (Charney and Devore 1979), and the possibility of resonant excitation of some large-scale flows by smaller scales (Shutts 1983; Hoskins et al. 1983) are probably the best known causes. Another dynamical explanation of weather regimes was presented by Marshall and Molteni (1993), which aims to determine a subspace of the model phase space spanned by the neutral vectors where multiple quasi-stationary solutions are likely to be located. In the case where there are two equally populated clusters and where the mean flow is located halfway between them, one neutral vector will be pointed toward them from the time mean flow. Unfortunately, this symmetry condition is unlikely to be satisfied in general, especially in the presence of forcing that biases the population of clusters and shifts the position of the mean flow. The Marshall and Molteni criterion is also not exclusive and there may be several neutral vectors, not all of which point toward multiple equilibria, among which one has to choose the correct one.

Recently Haines and Hannachi (1995; HH95 hereafter) developed a simple technique to look for possible quasi-stationary states from a GCM. This has been done by minimizing the streamfunction tendency based on the barotropic vorticity equation as in Branstator and Opsteegh (1989) and Anderson (1992) but within the EOF phase space of the model, thus reducing the number of degrees of freedom and allowing the algorithm to converge toward realistic solutions within the model attractor without being trapped by local minima or Rossby–Haurwitz waves. The ±PNA teleconnection patterns (Wallace and Gutzler 1981) were identified dynamically using this technique and also statistically using a probabilistic approach based on a Gaussian mixture distribution applied to the 10 yr of the GCM model trajectory.

The purpose of the study described in this manuscript is to extend the technique of HH95 to three dimensions by introducing a quasigeostrophic baroclinic vorticity equation as an extension to the simple barotropic vorticity equation and using a 3D phase space with two vertical levels. In HH95, only the streamfunction is defined within the EOF phase space but not the tendency, as it is derived from the barotropic vorticity equation. Here both *ψ* and its tendency are truncated to the same phase space. This truncation is a natural way to study the time evolution of the flow within the phase space and has the advantage of providing explicit information on the transition paths between regimes. The use of a baroclinic model was motivated by the works of Marshall and Molteni (1993) and Neven (1994), who all extended previous barotropic work on blocking to explore the baroclinic regime. We will review the technique of HH95 in section 2. In section 3, we describe the low-frequency variability of the model run using three-dimensional low-pass EOFs of the 500- and 250-mb levels. In section 4, we introduce a realistic, simplified two-level quasigeostrophic model to describe and diagnose the 3D quasi-stationary patterns. Section 5 presents the statistical analysis of the model trajectory projected onto the EOF phase space as in HH95 to identify model clusters describing statistical weather regimes. The role of transient eddies in determining the quasi-stationary states is introduced in section 6. Some results concerning the stability of the quasi-stationary patterns will be presented in section 7. Discussion of the results and conclusions will be presented in the last section.

## 2. Multiequilibria and the barotropic vorticity equation

The technique developed by HH95 is based on minimization of the streamfunction tendency of the barotropic vorticity equation to find the quasi-stationary states of the GCM, exploiting the reduced phase space of low-pass EOFs. For this purpose, EOFs are computed from the 500-mb streamfunction time series of a 10-yr perpetual January run of the UGAMP GCM with T42 horizontal resolution, 19 vertical levels, and fixed boundary conditions (see HH95 for more detail).

*ψ*

_{i}designates the

*i*th EOF and

*ψ̄*

*ψ*at each point within the

*n*-dimension EOF phase space isThe streamfunction tendency based on the barotropic vorticity equation iswhere

*ζ*

_{a}= ∇

^{2}

*ψ*+

*f*is the absolute vorticity.

*I,*whereover the area of concern

*A.*This technique is computationally cheaper than that of Branstator and Opsteegh (1989) or Anderson (1992), for example, because of the fewer degrees of freedom involved. It also converges toward a few realistic minima. HH95 applied it to the Pacific Ocean basin and, with modifications, over the entire Northern Hemisphere (NH) and obtained patterns showing the ±PNA teleconnection pattern associated, respectively, with ridges over the west coast and the central Pacific. Figure 1 shows the two solutions found by HH95, which were concentrated over the Pacific basin. The same solutions were also found by introducing the role of the eddies in the budget equation of the vorticity and by minimizing the resulting residual tendency (see HH95 for more details).

## 3. Low-frequency variability and vertical structure of the model

To extend the above results to include baroclinic dynamics, we consider the same perpetual January GCM integration used above. The dataset is sampled every 3/4 day to avoid aliasing of the diurnal cycle (Thuburn 1991). HH95 based their study on streamfunction at 500 mb, a level free of major topographic influence over the Pacific; we choose to focus in this manuscript on the levels 500 and 250 mb. We display in Fig. 2 the time-mean streamfunction field at both levels in the Northern Hemisphere. Both figures show two jets beginning over the east coast of the continents terminating the Pacific track with a weak ridge over the west of North America at 500 mb (Fig. 2a) and a more diffluent jet at 250 mb (Fig. 2b). The strength of the jet at 250 mb can be seen by comparison with 500 mb.

To look at the 3D structure of low-frequency anomalies, the streamfunction time series at 500 and 250 mb were filtered using a 5-day low-pass Lanczos filter (Duchon 1979). As the flow at the higher level is stronger, the streamfunction fields at each level were normalized by the square root of their respective spatial mean variance (rms). The rms of the streamfunction at 500 mb is 4.7 × 10^{7} m^{2} s^{−1} and 9.4 × 10^{7} m^{2} s^{−1} at 250 mb. Due to the different spacing of grid points at high and low latitudes on the model grid, a smooth weighing function that includes an area weighting is used when finding low-frequency EOFs over the Northern Hemisphere or over the Pacific basin. For the Pacific EOFs an additional weighting function has been used (see HH95) to concentrate over that basin and North America (20°–90°N, 130°E–80°W). EOFs over the NH have been computed (not shown). The first NH EOF explains around 12% of the total NH midlatitude variance and has some PNA elements but is dominantly hemispheric while EOF2 and EOF3 have a better PNA-like signature and show some modulation of the Pacific and Atlantic jets. The first three NH EOFs explain altogether around 27% of the total NH midlatitude horizontal and vertical variations.

HH95 concentrated on the Pacific sector to study weather regimes. As this manuscript is an extension to their work, we choose to concentrate on the same sector, so that comparisons can be made. A set of 5-day low-pass Lanczos filter EOFs over the Pacific are computed. Figure 3 shows the first three along with their respective percentage of variance (EVALUE). The first EOF explains about 17% and shows a modulation of the Pacific jet with an indication of a PNA signature. It is most similar to the barotropic EOF1 of HH95 but with some substantial differences. The Pacific high in EOF1 (Figs. 3a,b) has been shifted eastward in such a manner that the axis of the Pacific low–high centers becomes more meridional. EOF1 of HH95 showed a simple PNA structure with no modulation of the jet. EOF2, with 14%, still shows a modulation of the Pacific jet and bears some similarities to EOF2 of HH95. The third EOF (Figs. 3e,f) explains about 9% of the total low-pass variance and has smaller scales than in EOF1 and EOF2. At 500 mb, EOF3 has mainly a zonal orientation and bears some resemblance to EOF3 of HH95, while the same pattern at 250 mb is slightly different in that the positive anomaly has two high centers and the structure is less zonally oriented. Higher EOFs (not shown) tend to have smaller spatial scales with highs and lows at about the same latitude (zonal structure). The first three Pacific EOFs explain altogether about 40% of the total variance. The low-order phase space spanned by the previous low-pass EOFs constitutes the space in which we study the dynamical aspects of variability and regime structure in the next section.

## 4. Three-dimensional quasi-stationarity

### a. Theoretical frame

EOFs constitute a simple tool that provides the main directions of the low frequency variability. Consequently, they can capture and describe a whole set of low-frequency behavior of the atmospheric model. Furthermore, the first few EOFs generally point to the phase space direction of the composite flow patterns of the most frequently occurring persistent anomalies (Mo and Ghil 1987), which is probably due to the fact that EOFs generally point toward the largest concentrations of invariant measures in the phase space of the system. These concentrations may be the locus of some quasi-stationary states. As a result, the first few EOFs can have patterns similar to the most frequently occurring quasi-stationary events (Mo and Ghil 1987; HH95). These are some of the interesting properties of EOFs. Equally important is the orthogonality of these patterns, which permits the decomposition of any flow configuration within, and the projection of the model trajectory back onto, the EOF phase space. The tendency of the flow based, for example, on the prognostic potential vorticity equation is not confined to this phase space even if the flow is, but the tendency can still project onto it. It is this truncated tendency that is addressed here.

*ψ*

^{k}

_{i}

*i*th EOF at the

*k*th level (

*k*= 1 is 500 mb and

*k*= 2 is 250 mb), then the streamfunction field (

*ψ*

^{k}) at each point within the

*n*-dimensional 3D-EOF phase space iswhere

*X*

_{i}is the amplitude of the anomaly streamfunction for the

*i*th EOF. We consider as prognostic equations for the streamfunction tendency a two-layer quasigeostrophic model governed by the potential vorticity equations for 500 and 250 mb:where

*R*

_{k}is the internal Rossby deformation radius at level

*k.*As for the barotropic calculation in HH95 described above, we wish to find points of minimum streamfunction tendency within the EOF phase space. Departing from HH95 here, we demand that the tendency is also in the same phase space. In order to achieve this, the streamfunction fields expressed in Eq. (4) are written in vector form aswhere

__Ψ__

*ψ*

^{1},

*ψ*

^{2})

^{T}is the streamfunction field,

__Ψ̄__

*ψ̄*

^{1}

*ψ̄*

^{2}

^{T}is the mean flow, and

__Ψ__

_{i}*ψ*

^{1}

_{i}

*ψ*

^{2}

_{i}

^{T}is the

*i*th EOF. The superscript

*T*stands for the transpose operator. We denote by

__Ξ__

^{1},

*ξ*

^{2})

^{T}the quasigeostrophic potential vorticity vector and

__Ψ__

__Ξ__

__Ξ__

__Ψ__

*J*= (−

*J*

^{1}, −

*J*

^{2})

^{T}is the rhs vector of Eq. (5). Using Eq. (9) together with Eq. (7), one can obtain the projected streamfunction tendency onto the

*n*-dimensional EOF phase space, which can be written aswith

*X*

*X*

_{1}

*X*

_{n}

*F*

*X*

*F*

_{1}

*X*

*F*

_{n}

*X*

*F*

_{k}(

*X*) are given bywhere, in turn, 〈

__Ψ__

_{i}__Ψ__

_{j}*ψ*

^{1}

_{i}

*ψ*

^{1}

_{j}

*ψ*

^{2}

_{i}

*ψ*

^{2}

_{j}

Minimizing *n* is, without doubt, much easier than minimizing the tendency of the streamfunction (or potential vorticity) within the full phase space as in Branstator and Opsteegh (1989). The main advantage of this technique is that the entire phase space can be explored rather than just finding single local minima. It also allows consideration of models with more than one level, thus taking into account the vertical structure of the model. Another advantage of this extended technique, which is probably the most important here, is its ability to provide information about the flow directions between regimes in the phase space, as will be explained in section 7. This has not been possible in HH95.

This is because the right- and left-hand sides of the vorticity Eq. (2) (barotropic) or (9) (baroclinic), evolve into different spaces due to the nonlinearity of advection. Since the left-hand side is already in the EOF phase space, it is natural to obtain information about trajectories between regimes by projecting the right-hand side terms onto the same phase space.

### b. Quasi-stationary solutions

The Rossby deformation radius parameters *R*_{1} and *R*_{2} for the two levels 500 and 250 mb are considered, first, to be equal and fixed to 700 km. This value is appropriate for the 500–250-mb layer (see Marshall and Molteni 1993). These parameters and also the number of degrees of freedom of the system will be changed later to test the robustness of the solutions.

We started the minimization of *X*), while Fig. 4b shows the transformation of *X*) using an appropriate scaling function to show clearly the minima. The singular points of the dynamical system (10) within the phase plane of the first two Pacific EOFs are shown by dots (Fig. 4). The corresponding vector field *F*(*X*) indicating the nature of these singular points will be shown and discussed in section 7. Within the first three Pacific EOFs, the amplitudes and cost-function values of the solutions are shown in Table 1. The cost-function value of the mean flow is also shown. The differences in the *I* given by Eq. (3) (see Table 1 of HH95). Furthermore, the values of the cost-function at the minima are zero to machine accuracy, while in HH95 they were nonzero. These findings tell us that the minima found here are robust since they are significantly deeper than the climatology. For this reason, they are expected to be locations in the phase space where trajectories are slower than in other parts of the space and thus prominent on the statistics of the atmospheric model. From now on, all the figures that will be presented and discussed are, except when stated otherwise, from the three EOF phase space.

The solutions are presented in Figs. 5, 6, and 7. Figure 5 shows the anomaly solution 1 and the total flow of the solution 1 at 500 (Figs. 5a,b) and at 250 mb (Figs. 5c,d). This first minimum shows a Pacific blocking signature with strong amplitude. The high pressure center of the dipole is very close to the west coast of North America. The solution has a fairly barotropic signature. This solution is named block 1 (BL1). Solution 2 (Fig. 6), named block 2 (BL2), shows also a blocking over the Pacific with a different structure from block 1. The low pressure center of the dipole at 500 mb has shifted westward by about 20° (Fig. 6b). The vertical structure of the total flow (Figs. 6b,d) is not similar to the previous solution (Figs. 5b,d). In fact, the split at 250 mb (Fig. 6d) is not as strong as in block 1. BL1 and BL2 anomaly blocking solutions share some similarities with the PNA anomaly solution of HH95 (Fig. 1c). Note, for example, the same position of the flow ridge over the west coast in BL1 and BL2 (Figs. 5b,d and 6b) and in the PNA (Fig. 1d). However, the amplitudes in BL1 and BL2 are much stronger and the wave direction in these anomalies is parallel to the meridional direction (Figs. 5a,c and 6a,c), while it is southwest–northeast oriented in the PNA solution (Fig. 1c). The third minimum (Fig. 7) shows a fairly zonal flow over the Pacific. The jet over the east Pacific at 500 mb (Fig. 7b) is slightly stronger than at 250 mb (Fig. 7d). We choose to call it zonal flow (ZL). The ZL anomaly solution (Figs. 7a,c) bears some resemblance to the −PNA anomaly pattern of HH95 (Fig. 1a). Charney and Devore’s (1979) equilibrium states were of a similar type: one with high index equivalent to a blocking flow regime and the other with a low index or zonal flow. The solutions obtained from the truncated barotropic streamfunction tendency (not shown) are similar to BL1 and BL2. Furthermore, only a ZL-type solution is found when the quasigeostrophic streamfunction tendency is not projected onto the EOF phase space. One concludes that both the vertical structure and the tendency truncation contribute to changes in the solutions.

To examine the influence of the parameter changes on the solutions, the number of degrees of freedom has been increased from two and three up to 10 EOFs with no major changes in the solutions. These solutions are more robust to changes in EOF number than those of HH95, where it is found that for nine and more EOFs, one of the solutions (+PNA) ceased to be quasi-stationary. Although the amplitudes change because of the newly added EOFs, the spatial structure of the solutions remain almost unchanged. Table 2 shows the different solutions for four and five EOFs. It can be seen that if the contribution of the newly added EOF is not strong for one solution, then its amplitudes may not change too much (e.g., BL2 at three and four EOFs). If this contribution is strong, then there may be a change in the coefficient amplitudes of the solution (BL1 at three and four EOFs). The values of the cost-function at the minima have also changed at five dimensions for BL2 and ZL but are still at least 50 times smaller than for the climatology (e.g., ZL solution) meaning that the solutions are still significant. One might expect that the cost-function can only get smaller as the number of degrees of freedom increases as stationary Rossby–Haurwitz solutions can be attained. But we have to bear in mind that the minimized quantity is not the full norm of the tendency (as it was in HH95); rather it is the norm of its projection onto the phase space. Thus, when a new degree of freedom is added the tendency vector changes (and also its norm), and in particular it may cease to become strictly orthogonal to the phase space. Thus, while untruncated tendencies will become smaller as more degrees of freedom are added, the truncated tendencies are not necessarily monotonically decreasing. For example, the cost-function minima increase in value as the fourth and fifth degrees of freedom are added (BL2 and ZL solutions) but then decrease for six and more degrees of freedom.

To further test the robustness of our technique, the sensitivity of the solutions to the Rossby deformation radii is considered. We set the lower parameter *R*_{1} to 700 km and the upper parameter *R*_{2} to 500 km, values used by Vautard and Legras (1988) to look for weather regimes from a quasigeostrophic channel model forced by a local baroclinic jet. These values were taken from the U.S. standard atmosphere (Berry et al. 1973). With the new values of *R*_{1} and *R*_{2}, we found almost the same solutions.

Next, we see if the dynamical states found in this section influence GCM trajectories.

## 5. Statistical weather regimes

To study the behavior of the temporal trajectory, the streamfunction time series at 500 and 250 mb was projected onto the phase space spanned by the leading 3D EOFs and sampled every 3/4 day from the model run. As in HH95, we want to verify if the 3D dynamical weather regimes found previously do influence the model trajectory by looking at the phase space distribution of the sampled time trajectory. The problem of looking for clusters within a given dataset is well known to be a difficult one. Numerous methods and techniques have been proposed to solve it, including hierarchical techniques and aggregation around moving centers. This classification problem originated initially from the field of pattern recognition and, as pointed out by Aart and Korst (1990), constitutes a class of problems that can often be easily solved by human beings but are very hard to solve by computers. During the last decade or so, attention has been directed toward the use of probabilistic techniques and the estimation and use of probability density functions (Silverman 1986). This approach has been applied to the atmosphere (Hansen 1986; Hansen and Sutera 1986; Molteni et al. 1988). On the other hand, it has also been shown that the pdf is generally unstable to small changes of its parameters unless very long datasets are used (Nitsche et al. 1994).

*n*-dimensional EOF phase space is assumed to have a distribution function

*f*(

*x**r*-multinomial distributions

*f*

_{m}

__);__

*x**f*

_{m}is supposed to represent the

*m*th cluster with center

*a*_{m}

**C**

_{(m)}. It is given bywhere

*x**x*

_{1},

*x*

_{2}, . . .

*x*

_{n}) is a point in the

*n*-dimensional EOF phase space,

*r*is the number of clusters forming the whole population, and

*λ*

_{m}is the proportion of the

*m*th cluster satisfying the constraint

*L*

_{r}over all

*N*observed states:

This technique ought to be useful for many reasons including its simplicity, the possibility of testing the results by comparison to alternatives, and the fact that the method can reject unwanted classes by setting their proportions, *λ*_{m}, to zero.

**C**

_{(m)}, and then the unknowns

*a*_{m}

*λ*

_{m}are sought by minimizing −Log(

*L*

_{r}). This choice prevents the generation of singularities of the likelihood during the minimization process (for more details see Everitt and Hand 1981). Since the centroids of the different clusters are expected not to be too different from the centroid of the whole population, the covariances

**C**

_{(k)}were chosen to be of the form

**C**

_{(k)}

*α*

*k,*

*α*is a real parameter between 0 and 1, and Σ is the covariance matrix of the entire population. Many experiments have been made by varying the parameter

*α*. The solutions were found to vary very slightly. For each experiment, where the dimension

*n*of the EOF phase space and the number

*r*of classes are fixed, a hundred random initial conditions are generated. The final solutions are then selected among the set of all the minima according to their likelihood. The case

*n*= 3 and

*α*= 0.7 are considered here for discussion.

The number of classes has been varied through the experiment. With two clusters, the most likely solutions contain one pattern with small amplitudes close to the origin (mean flow) and a second pattern looking like one of the dynamical solutions mentioned before (BL1, BL2, or ZL) but with smaller amplitudes. When three classes are introduced, the most likely solution has three patterns similar to the dynamical solutions although smaller in EOF amplitudes. Table 3 shows the pattern amplitudes and the likelihoods. The likelihood of the mean flow (one class) is also shown. Figures 8, 9, and 10 show the solution patterns presented in Table 3. Although the amplitudes are smaller than the dynamical quasi-stationary states, the anomaly patterns look fairly similar (Figs. 5, 8; 6, 9; and 7, 10).

*χ*

^{2}, which gives a measure for this test, is used here because it can explicitly be approximated by

*χ*

^{2}

_{dr}

*L*

_{1}

*L*

_{r}

*r*is the number of clusters (two or three);

*d*

_{r}, the number of degrees of freedom of the

*χ*

^{2}, is equal to the difference in the number of parameters in the two hypothesis;

*L*

_{r}is the likelihood of the solution with

*r*classes; and

*L*

_{1}is the likelihood of one cluster with covariance Σ, centered at the climatology (see Wolfe 1970 for other possible approximations). The largest likelihood in the two classes type

*L*

_{2}is about 2503. From Table 3, it can be seen that the two-cluster solution is automatically rejected and the three-cluster type can be accepted. Table 3 shows also the case with a four-cluster model, which is more likely than the three-cluster model and where the first three clusters (not shown) are similar to BL1, BL2, and ZL, respectively. The fourth cluster has weak amplitudes and is very close to the mean flow. If the number of classes is increased, some of the previous clusters split to produce two fairly similar clusters and the proportions of the newly found clusters become so small that their significance becomes questionable. One concludes that the dataset can be accepted with three significant clusters whose centers are shown in Figs. 8, 9, and 10.

## 6. Weather regimes and transient eddies

Although large-scale weather regimes are generally accepted as quasi-stationary or “persistent,” the transients due to small scales remain generally active. This is particularly important for regimes like blocking where the role of transients is crucial (Shutts 1983; Illari 1984). In this section we will consider the role of eddies in determining weather regimes.

*J*

*ψ*

^{k}

*ψ*

^{k}

_{d}

*ξ*

^{k}

*f*

*ξ*

^{k}

_{d}

*J*

*ψ*

^{k}

*ξ*

^{k}

*f*

*J*(

*ψ*′

^{k},

*ξ*′

^{k})

*k*= 1, 2,

__Ψ__

_{d}*ψ*

^{1}

_{d}

*ψ*

^{2}

_{d}

^{T}is the streamfunction displacement (within the EOF phase space) from the climatology

__Ψ̄__

__Ψ__

_{d}^{n}

_{i=1}

*X*

_{i}

__Ψ__

_{i}*ξ*

^{1}

_{d}

*ξ*

^{2}

_{d}

^{T}=

__Ξ__

_{d}__Ψ__

_{d}^{−1}.

^{−1}

_{1},

_{2}), the residual tendencies, are given by rearranging Eq. (19):Approximate solutions to Eq. (20) can be given by minimizing the appropriate norm of its left-hand side. This is the technique applied in HH95. Within the present context, however, one is looking for approximate solutions that minimize the lhs of Eq. (20) within the phase space. These are obtained by minimizing the projection of

^{−1}(

__)__I

*X*) = [

_{1}(

*X*), . . .,

_{n}(

*X*)] is defined byAnother simple way to understand Eq. (22) without using Eq. (19) is to go back to the expression of

*F*(

*X*) in Eq. (11). In this equation, each component

*J*

^{k}(

*k*=1, 2) of the tendency vector

T

*J*

^{k}

*J*

*ψ*

^{k}

*ξ*

^{k}

*f*

*J*

*ψ*

^{k}

*ξ*

^{k}

_{d}

*J*

*ψ*

^{k}

_{d}

*ξ*

^{k}

*f*

*ξ*

^{k}

_{d}

I

Here, _{d}(*X*) has been evaluated for states over the Pacific basin and three minima were found. Table 4 shows the amplitude of solutions with their cost-functions within the phase space of the first three EOFs. The solutions are in the commonly observed range and relatively weaker than the solutions obtained in section 4. The first two solutions (Figs. 11 and 12) show, respectively, and relative to the mid-Pacific, an eastward and a center-shifted Pacific block. The third anomaly solution (Fig. 13) has weak wave amplitudes and is closer to the origin. It is fairly similar to the previous dynamical and statistical zonal solutions, although with weaker amplitudes. The shift between the two blocks of Figs. 11 and 12 is clearer than the shift between the blocks of Figs. 5 and 6. Also, the central Pacific block (Fig. 12) is shifted slightly southward. The transient eddies have mainly reduced the anomaly amplitudes of the solutions, which is in agreement with the argument stated above about finding small time-mean perturbations that have the same effect as the transients, making the statistical patterns closer to these solutions than to the dynamical solutions of section 4. In particular, the statistical and dynamical solution 2 (Figs. 9 and 12) both look much like the −PNA solution (Figs. 1a,b).

## 7. Stability of quasi-stationary states

The methodology adopted here concerning the search for quasi-stationarity through the use of a dynamical system is helpful in studying the stability of its singular points. In fact, consideration of weather regime stability may shed some light on their mutual transitions, which are crucial in determining, for example, their onsets and/or breakup. This representation also allows the determination of the transition times between regimes. This is illustrated in Fig. 14. The arrows in this figure indicate the projected streamfunction tendency *F*(*X*) = [*F*_{1}(*X*), *F*_{2}(*X*)] within the phase plane of the first two EOFs as calculated from (11). It shows three types of behavior and the possible transitions between these states can be noted. The solution block 1 (BL1) behaves as an unstable node, while block 2 (BL2) is a saddle point. The third zonal solution (ZL) is a stable node. For a given transition path from one regime to another, one can find and follow several configurations of the flow, and this helps understand the transition process within the model phase space. These transitions between regimes, generally associated with the nonlinearity of the atmospheric system and not requiring changes to external forcing, have small atmospheric predictability (Palmer and Anderson 1994). Understanding and following the transition paths, such as those shown in Fig. 14, can address this predictability problem. Notice in particular the agreement between the unstable nature of blocking and the character of BL1 and BL2 (Fig. 14). It is, in fact, generally accepted that blocking tends to be highly unstable, which prevents forecasters from making reliable forecasts due to its abrupt onset/breakdown (Kimoto et al. 1992; Anderson 1992). The unstable character of blocking caught in such a low-dimensional phase space is a result of a “competition” for stability between zonal and blocking regimes.

At three and higher dimensions the stability of the quasi-stationary states can be studied directly by linearizing system (10) around these points or by integrating the full nonlinear dynamical system forward in time. For example, within the phase space of the first three EOFs, system (10) has been integrated from different initial conditions using a finite difference scheme. Figures 15a and 15b show the result of this integration for two initial conditions chosen near ZL, and Figs. 15c and 15d show an integration beginning near BL2. These integrations illustrate stability characteristics similar to those seen in Fig. 14. Notice that the trajectory leaves BL2 and heads toward ZL very slowly (initial flatness of the trajectory) at the beginning (Fig. 15c) due to its saddle nature, while it quickly leaves the unstable BL1 focus (not shown). If the initial condition is chosen randomly from the neighborhood of BL1, the probability of convergence toward BL2 is much smaller compared to the probability of convergence toward ZL due to the saddle nature of BL2. One possible picture of blocking decay is then a direct transition to a zonal flow. The convergence from BL1 to BL2 is possible through the stable manifold of BL2. It is generally difficult to get to BL2 through this stable manifold due to numerical errors. In practice, however, this can be done by linearizing system (10) around BL2, applying an infinitesimal displacement to BL2 along the stable eigenvector of the linearized system, and then integrating system (10) backward in time from this perturbed initial condition in order to get to BL1. The obtained trajectory can then be approximately identified as the stable manifold of BL2.

The time integrations performed above are of great interest since one is able to get an approximation of the time needed for a transition between one regime and another to be accomplished. The transition times between regimes are very important for forecasters. They have been calculated within the phase space of the three EOFs. The transition time between BL1 and ZL is around 26 days, while it is around 37 days between BL2 and ZL. These values have been calculated after integration of system (10) forward in time starting from two initial conditions near BL2 (Figs. 15c,d) and BL1 (not shown), respectively. Due to the saddle nature of BL2, we performed a backward integration of Eq. (10) beginning near BL2 in order to get to BL1, and we find 33 days as transition time from BL1 to BL2. One should remember that these transition times are not absolute but depend, of course, on the particular perturbation superimposed upon the regime (see Fig. 14). These relatively high values of transition times are mainly a consequence of the averaging effect since Fig. 14 represents the system behavior in an “average” sense.

The modon approach to blocking, which can be extended to the quasigeostrophic framework (Haines and Marshall 1987), shows that once a block is formed it may start moving upstream due to its dipolar nonlinear structure. Although the previous transitions can be accomplished only in the presence of special perturbations, and since the blocking solution BL2 is shifted westward with respect to BL1, the transition from BL1 to BL2 reminds us of the upstream movement of modons. The vertical structure of the blocking solution BL2 shows a weak upper dipolar structure (at 250 mb), suggesting another, although less likely, picture of blocking decay. When a block breaks down, it starts losing its upper dipole and eventually dissolves into a zonal flow.

Charney and Devore’s (1979) regimes obtained through nonlinear interaction between the externally driven zonal flow and Rossby waves forced by surface topography were stable. This led them to hypothesize the existence of unresolved small perturbations that render these states unstable, hence allowing transitions between them. Although this has been challenged later by Mukougawa (1987) within a linear framework, it is still probable that small scales might introduce instabilities to the stationary states. In particular, the zonal flow may have unstable directions following higher EOF directions, making the transitions to and from it possible through the unstable and stable manifolds, respectively. In fact, the zonal solution, which is a stable node at two and three dimensions of the phase space, has an unstable manifold at four dimensions. This also happens even at three dimensions when the role of the eddies is included (Table 4), where the zonal solution has lost its stability and the transition between ZL to BL1 becomes possible.

## 8. Discussion and conclusions

In this manuscript we have addressed the problem of the existence of large-scale three-dimensional weather regimes in the northern extratropical atmosphere from a GCM with a focus on the Pacific Ocean. We have extended the technique developed by HH95 to the 500- and 250-mb levels. HH95 minimized the streamfunction tendency within the EOF phase space onto which an important part of the model attractor can be projected. Here, the streamfunction tendency, calculated from the quasigeostrophic potential vorticity equation, is projected onto the phase space of three-dimensional EOFs in order to identify locations of singular points of the resulting dynamical system.

These singular points were found using the Pacific EOF phase space and had almost zero projected streamfunction tendency even when compared to the mean flow projected tendency. This means that the near-zero tendencies are not only due to the projection effect. This technique has the advantage of being able to explore the entire phase space and allows consideration of models with multilevels. It also allows a direct exploration of the flow tendency and helps the study of the stability characteristics and their consequences on the regimes within the phase space. The solutions primarily have blocking and nonblocking (zonal) structures. These solutions have been shown to be quite robust to changes in number of EOFs retained. Apart from slight differences in amplitudes, the solutions are found to compare to those of HH95. Solutions computed from the truncated barotropic streamfunction and the nontruncated quasigeostrophic streamfunction tendency (two levels) suggest that differences with HH95 are attributed to both the vertical structure and the truncation of the flow tendency to the basis of the model.

A statistical analysis of the GCM trajectory projected onto the same EOF phase space reveals that these dynamical states can influence the trajectory. Pdfs have shown that there are three primary clusters in the GCM trajectory with centers similar to the dynamical ones, although with smaller EOF amplitudes. Similar solutions are also obtained when the observed statistics of transient fluxes are used. The obtained blocking solutions are more clearly separated than in the previous case (with no eddy contributions). In fact, the first blocking solution is located near the West Coast, while the second is clearly shifted further west in the central Pacific. This may correspond to two phases for Pacific blocking.

The quasi-stationary states we have found are based on a search for minimum tendencies, while our statistical approach may produce states that are primarily recurrent. The difference between these two types of states has been noted recently by Michelangeli et al. (1995) by studying the difference between stationarity and recurrency. They found that the quasi-stationary patterns differ significantly from the recurrent ones. They argued that these latter have a systematic slow evolution explaining this difference. However, we still think that recurrent and quasi-stationary patterns should not differ too much (see Mukougawa 1988), and we believe that the mismatch in amplitudes between the singular fixed points and our cluster centroids may be due to the lack of dissipation in the quasigeostrophic model.

As stated above, results obtained respectively from HH95, where ±PNA were obtained at 500 mb, and results from the present manuscript suggest that both vertical structure and tendency truncation to the basis of the model, are necessary to obtain blocking over the Pacific. The tendency truncation, like any other truncation (spectral for example), is a natural way to make the streamfunction evolution within the model basis meaningful. Similarities between the blocking (and also zonal) regimes and the ±PNA patterns (e.g., HH95) is well known (Frederiksen 1992). We carried out experiments to look for quasi-stationary states of Eqs. (5) and (6), which omit the first EOF, which contains the PNA pattern. Such experiments failed to identify any blocking regimes.

In connection with the dynamics of these regimes, the stability of the quasi-stationary states has been studied within the EOF phase space by integrating the corresponding dynamical system forward in time starting from a point within a neighborhood of the singular points of the system. This gives a picture of what the transitions between these patterns might look like in reality. In particular, the ZL regime is found to be the most stable regime with more frequent transitions toward it. The unstable or saddle character of the blocking regimes is in agreement with their unpredictable nature. The blocking regime BL1 can degenerate directly into a ZL flow or, alternatively, into a blocking type BL2, depending on the initial perturbation imposed upon it, and then eventually into a ZL flow. The possible transition between the two blocking states bears some similarities with the upstream movement of modons. The introduction of transients have changed the stability of the zonal regime making the hypothesis of a transition from zonal to blocking flows plausible. Transition times between regimes have also been found, and it is shown that one needs around a month on average to complete a transition. Here, transition begins at the time when the first regime starts decaying and ends when the new regime is well established. This and the averaging effect explain why these periods are longer than onset times for blocking. One might expect that the application of the above technique to real observations may shed further light on the characteristics of weather regimes. Of particular interest would be transition diagrams and the transition times between regimes, which can be measured and compared with those found above.

## Acknowledgments

This work was funded by the NERC under Grant GR3/8274. The 10-yr run of the GCM was supported by UGAMP. I am grateful to Dr. K. Haines for helpful comments and discussions. Thanks are also due to Dr. G. Branstator for his thorough and constructive review and to an anonymous reviewer for helpful comments.

## REFERENCES

Aart, E., and J. Korst, 1990:

*Simulated Annealing and Boltzmann Machine.*John Wiley and Sons, 272 pp.Anderson, J. L., 1992: Barotropic stationary states and persistent anomalies in the atmosphere.

*J. Atmos. Sci.,***49,**1709–1722.Barnston, A. G., and R. E. Livezey, 1987: Classification, seasonality, and persistence of low-frequency atmospheric circulation patterns.

*Mon. Wea. Rev.,***115,**1083–1126.Berry, F. A., E. Bollay, and N. R. Beers, 1973:

*Handbook of Meteorology.*McGraw-Hill, 1068 pp.Branstator, G., 1987: A striking example of the atmosphere’s leading travelling pattern.

*J. Atmos. Sci.,***44,**2310–2323.——, and J. D. Opsteegh, 1989: Free solutions of the barotropic vorticity equation.

*J. Atmos. Sci.,***46,**1799–1814.Charney, J. G., and J. G. Devore, 1979: Multiple flow equilibria in the atmosphere and blocking.

*J. Atmos. Sci.,***36,**1205–1216.Dole, R. M., 1986: Persistent anomalies of the extratropical Northern Hemisphere wintertime circulation: Structure.

*Mon. Wea. Rev.,***114,**178–207.——, and D. N. Gordon, 1983: Persistent anomalies of the extratropical Northern Hemisphere winter circulation: Geophysical distribution and regional persistent characteristics.

*Mon. Wea. Rev.,***111,**1567–1586.Duchon, C. E., 1979: Lanczos filtering in one and two dimensions.

*J. Appl. Meteor.,***18,**1016–1022.Everitt, B. S., and D. J. Hand, 1981:

*Finite Mixture Analysis.*Chapmann and Hall, 143 pp.Frederiksen, J. S., 1992: Towards a unified instability theory of large-scale atmospheric disturbances.

*Trends Atmos. Sci.,***1,**239–261.Haines, K., and J. Marshall, 1987: Eddy-forced coherent structures as a prototype of atmospheric blocking.

*Quart. J. Roy. Meteor. Soc.,***113,**681–704.——, and A. Hannachi, 1995: Weather regimes in the Pacific from a GCM.

*J. Atmos. Sci.,***52,**2444–2462.Haltiner, J. G., and R. T. Williams, 1980:

*Numerical Prediction and Dynamic Meteorology.*John Wiley and Sons, 477 pp.Hannachi, A., and B. Legras, 1995: Simulated annealing and weather regimes classification.

*Tellus,***47A,**955–973.Hansen, A. R., 1986: Observational characteristics of atmospheric planetary waves with bimodal amplitude distributions.

*Advances in Geophysics,*Vol. 29, Academic Press, 101–133.——, and A. Sutera, 1986: On the probability density distribution of large-scale atmospheric wave amplitude.

*J. Atmos. Sci.,***43,**3250–3265.Horel, J. D., 1981: A rotated principal component analysis of the interannual variability of the Northern Hemisphere 500 mb height field.

*Mon. Wea. Rev.,***109,**2080–2092.Hoskins, B. J., and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing.

*J. Atmos. Sci.,***38,**1179–1196.——, I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems.

*J. Atmos. Sci.,***40,**1595–1612.Illari, L., 1984: A diagnostic study of the potential vorticity in a warm blocking anticyclone.

*J. Atmos. Sci.,***41,**3518–3526.Kimoto, M., M. Mukougawa, and S. Yoden, 1992: Medium-range forecast skill variation and blocking transition: A case study.

*Mon. Wea. Rev.,***120,**1616–1627.Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking and variations in atmospheric predictability.

*J. Atmos. Sci.,***42,**433–471.Lichtenberg, A. J., and M. A. Lieberman, 1983:

*Regular and Stochastic Motion.*Springer-Verlag, 499 pp.Lorenz, E. N., 1963: Deterministic nonperiodic flow.

*J. Atmos. Sci.,***20,**130–141.——, 1965: A study of the predictability of a 28-variable atmospheric model.

*Tellus,***17,**321–333.Marshall, J. C., and F. Molteni, 1993: Toward a dynamical understanding of planetary-scale flow regimes.

*J. Atmos. Sci.,***50,**1792–1818.Michelangeli, P. A., R. Vautard, and B. Legras, 1995: Weather regimes: Recurrence and quasi-stationarity.

*J. Atmos. Sci.,***52,**1237–1256.Mo, K. C., and M. Ghil, 1987: Statistics and dynamics of persistent anomalies.

*J. Atmos. Sci.,***44,**877–901.——, and ——, 1988: Cluster analysis of multiple planetary flow regimes.

*J. Geophys. Res.,***93,**927–952.Molteni, F., S. Sutera, and N. Tronci, 1988: The EOFs of the geopotential eddies at 500 mb in winter and their probability density distributions.

*J. Atmos. Sci.,***45,**3063–3080.——, S. Tibaldi, and T. N. Palmer, 1990: Regimes in the wintertime circulation over northern extratropics. I: Observational evidence.

*Quart. J. Roy. Meteor. Soc.,***116,**31–67.Mukougawa, H., 1987: Instability of topographically forced Rossby waves in a two layer model.

*J. Meteor. Soc. Japan,***65,**13–25.——, 1988: A dynamical model of “quasi-stationary” states in large-scale atmospheric motions.

*J. Atmos. Sci.,***45,**2868–2888.Namias, J., 1950: The index cycle and its role in the general circulation.

*J. Meteor.,***7,**130–139.——, 1964: Seasonal persistence and recurrence of European blocking during 1958–1960.

*Tellus,***16,**94–407.Neven, E. C., 1994: Baroclinic modons on a sphere.

*J. Atmos. Sci.,***51,**1447–1464.Nitsche, G., J. M. Wallace, and C. Kooperberg, 1994: Is there evidence of multiple equilibria in planetary wave amplitude statistics?

*J. Atmos. Sci.,***51,**314–322.Palmer, T. N., 1988: Medium and extended range predictability and stability of the Pacific/North American mode.

*Quart. J. Roy. Meteor. Soc.,***114,**691–713.——, and D. L. T. Anderson, 1994: The prospects for seasonal forecasting—A review paper.

*Quart. J. Roy. Meteor. Soc.,***120,**755–793.Rex, D., 1950: Blocking action in the middle troposphere and its effect upon regional climate.

*Tellus,***2,**196–211.Shutts, G. J., 1983: The propagation of eddies in diffluent jet streams: Eddy forcing of blocking flow fields.

*Quart. J. Roy. Meteor. Soc.,***109,**737–761.Silverman, B. W, 1986:

*Density Estimation for Statistics and Data Analysis.*Chapman and Hall, 175 pp.Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns.

*J. Atmos. Sci.,***40,**1363–1392.Thuburn, J., 1991: Data sampling strategies for general circulation models.

*Quart. J. Roy. Meteor. Soc.,***117,**385–397.Tung, K. K., and A. J. Rosenthal, 1985: Theories of multiple equilibria—A critical reexamination. Part I: Barotropic models.

*J. Atmos. Sci.,***42,**2804–2819.Vautard, R., 1990: Multiple weather regimes over the North Atlantic: Analysis of precursors and successors.

*Mon. Wea. Rev.,***118,**2056–2081.——, and B. Legras, 1988: On the source of low frequency variability. Part II: Nonlinear equilibration of weather regimes.

*J. Atmos. Sci.,***45,**2845–2867.Wallace, J. M., and D. S. Gutzler, 1981: Teleconnections in the geopotential height field during the Northern Hemisphere winter.

*Mon. Wea. Rev.,***109,**784–812.Wiin-Nielsen, A., 1979: Steady states and stability properties of a low order barotropic system with forcing and dissipation.

*Tellus,***31,**375–386.Wolfe, J. H., 1970: Pattern clustering by multivariate mixture analysis.

*Multiv. Behav. Res.,***5,**329–350.

Solutions that minimize the projected streamfunction tendency onto the low-pass Pacific EOF phase space (three EOFs): block 1 (BL1), block 2 (BL2), and zonal (ZL) (units of T : 1.6 × 10^{−7} s^{−2}).

As in Table 1 but using the phase space of the first four and five EOFs (same units as in Table 1).

Mixture analysis clusters (three- and four-class models) with the corresponding log likelihood (**C*** _{(k)}* = 0.7Σ). The single-class model for the climatology (

**C**

_{(1)}= Σ) is also shown. The likelihood of the most likely solution with a two-class model is 2503.

As in Table 1 but for the projected streamfunction tendency T _{d} onto the phase space of the first three EOFs when the role of transient eddies is included (units as in Table 1).