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  • View in gallery

    T21 orography of the NH. Contour interval is 500 m; zero contour omitted.

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    Wintertime (DJF) extratropical nonzonal diabatic heating at (left) 300 and (right) 700 hPa derived from NCEP–NCAR reanalysis data and used in the QG three-level model. See text for details.

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    Modeled and observed time-mean zonal wind profiles for the three model levels and corresponding vertical layers, respectively.

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    (left) Mean geopotential height of the (top) 167-, (middle) 500-, and (bottom) 833-hPa model level. (right) Geopotential height standard deviation of the (top) 167-, (middle) 500-, and (bottom) 833-hPa model level.

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    (left) First and (right) second EOF of 833-hPa geopotential height.

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    Power density spectrum of the (left) first and (right) second PC of the 833-hPa geopotential height.

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    Estimated probability density in the space spanned by the first two PCs of the 833-hPa geopotential height. Contour interval is 0.01; zero contour omitted. The thick lines enclose areas with significantly higher probability than to be expected in the case of the adapted red noise process (at the 95% level), and thus mark the circulation regimes. The regimes are denoted as indicated.

  • View in gallery

    Composite anomaly pattern of the (left) NAO regime and the (right) AO+ regime at the 833-hPa level.

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    Zonal wind profiles of a typical steady-state example.

  • View in gallery

    First EOF of the 833-hPa geopotential height for experiments (upper left) A, (upper right) B, (lower left) C,and (lower right) D. For a better comparison of pattern structures, individual contour intervals are used for each pattern.

  • View in gallery

    The first 5000 days of the first PC of 833-hPa geopotential height for experiments (upper left) A, (upper right) B, (lower left) C, and (lower right) D.

  • View in gallery

    Estimated probability density in the space spanned by the first two PCs of 833-hPa geopotential height for (upper left) experiment A, (upper right) experiment B, (lower left) experiment C, and (lower right) experiment D. Contour interval is 0.01; zero contour omitted. The thick lines enclose areas with unexpectedly high probability, as in Fig. 7.

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    The bifurcation route to regime behavior hypothesized to be common among realistic atmospheric models.

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Toward Understanding the Dynamical Origin of Atmospheric Regime Behavior in a Baroclinic Model

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  • 1 Alfred Wegener Institute for Polar and Marine Research, Potsdam, Germany
  • | 2 Department of Geophysics, Faculty of Physics and Mathematics, University of Concepción, Concepción, Chile
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Abstract

Dynamical mechanisms of atmospheric regime behavior are investigated in the context of a quasigeostrophic three-level T21 model of the wintertime atmospheric circulation over the Northern Hemisphere. The model, driven by realistic orography and using a thermal forcing determined by a newly developed tuning procedure, is shown to possess a reasonable climatology and to simulate the Arctic Oscillation quite realistically. It exhibits pronounced internally generated interannual and decadal variability and, in particular, circulation regimes that agree fairly well with observed ones. Two known hypotheses about the origin of regime behavior, as it occurs in the model herein are addressed: (i) multiple equilibria and (ii) chaotic itinerancy between attractor ruins. The first hypothesis is falsified at very high probability, while the second is likely to be true.

* Current affiliation: Max-Planck-Institut für Plasmaphysik, Garching, Germany

+ On leave from A. M. Obukhov Institute of Atmospheric Physics, Moscow, Russia

Corresponding author address: Mario Sempf, Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany. Email: msempf@ipp.mpg.de

Abstract

Dynamical mechanisms of atmospheric regime behavior are investigated in the context of a quasigeostrophic three-level T21 model of the wintertime atmospheric circulation over the Northern Hemisphere. The model, driven by realistic orography and using a thermal forcing determined by a newly developed tuning procedure, is shown to possess a reasonable climatology and to simulate the Arctic Oscillation quite realistically. It exhibits pronounced internally generated interannual and decadal variability and, in particular, circulation regimes that agree fairly well with observed ones. Two known hypotheses about the origin of regime behavior, as it occurs in the model herein are addressed: (i) multiple equilibria and (ii) chaotic itinerancy between attractor ruins. The first hypothesis is falsified at very high probability, while the second is likely to be true.

* Current affiliation: Max-Planck-Institut für Plasmaphysik, Garching, Germany

+ On leave from A. M. Obukhov Institute of Atmospheric Physics, Moscow, Russia

Corresponding author address: Mario Sempf, Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany. Email: msempf@ipp.mpg.de

1. Introduction

Hints on the existence of preferred quasi-stationary configurations of the planetary-scale atmospheric flow, referred to as circulation regimes, have been known for a number of decades (Rossby 1939; Baur 1947; Namias 1950; Rex 1950a, b). When extensive instrumental records had become available, numerous studies were devoted to the detection of circulation regimes over the Northern Hemisphere (NH), mostly for wintertime, by using sophisticated statistical methods. The results obtained became increasingly similar, see, for example, Kimoto and Ghil (1993a), Michelangeli et al. (1995), Corti et al. (1999), Monahan et al. (2001), and many others. However, the dynamical mechanisms underlying these regimes, and driving the transitions between them, are still not completely understood, leaving an important, unresolved question for climate science.

Owing to the irregularity of the time intervals between regime transitions, the atmospheric regime behavior substantially contributes to natural fluctuations in the climate system on interannual and decadal time scales (Rossby and Willet 1948; Rossby 1959; Palmer 1998). Furthermore, it is assumed that anthropogenic climate change becomes manifest in a change of the frequency of occurrence rather than in a metamorphosis of the structure of the natural regimes (Palmer 1998, 1999; Corti et al. 1999). Therefore, understanding the mechanisms of atmospheric regime behavior could be a key to a better understanding of climate variability and change.

One well-known concept of explaining regime behavior is the so-called multiple equilibria theory introduced by Charney and DeVore (1979) and refined by Charney and Straus (1980) and Charney et al. (1981). In this theory, regimes are assumed to correspond to stationary solutions (steady states, fixed points) of the equations governing the large-scale atmospheric flow. In a strongly simplified barotropic channel model of midlatitude flow over orography, Charney and DeVore (1979) analytically proved the existence of three equilibrium solutions, two of them being stable. One of these point attractors corresponds to a high-index state, that is, a state with strong zonal flow (cf. Rossby 1939), and the other one to a low-index state with weakened zonal flow and large wave amplitudes. These fixed points were considered as being not entirely stable, but unstable with respect to perturbations on smaller scales not resolved by the model. Such instabilities are necessary for regime transitions to occur. Many studies were inspired by the multiple equilibria theory. Källén (1981) found a high-index state and a low-index state, both stable, in a highly simplified barotropic model with spherical geometry. In other studies, such as Yoden (1985), Legras and Ghil (1985), De Swart (1989), Shil’nikov et al. (1995), and Crommelin et al. (2004), numerical bifurcation analyses of steady states were conducted for similarly simple or slightly more complex models, and routes to chaos were examined. In the case of chaotic model dynamics, it is particularly interesting whether regime behavior emerges and whether the regimes are situated in the neighborhood of steady states in the model’s phase space, as to be expected due to the slowing down of dynamics near fixed points. This has been shown to be true in a few investigations of barotropic models, for example, by Legras and Ghil (1985) and Crommelin (2003). Notably, in the latter work a model with spectral T21 resolution (thus resolving the feedback between planetary and synoptic scales) and with realistic orography of the NH has been used. This model possesses two domains in phase space with particularly high probability of being visited by trajectories, corresponding to a high-index regime and a low-index regime. A weakly unstable fixed point is embedded in the high-index regime, whereas three weakly unstable fixed points are embedded within the low-index regime. Thus the validity of the multiple equilibria theory has been demonstrated for a comparatively realistic barotropic model.

Not only the structures of the regimes themselves, but also the mechanisms that lead to transitions between regimes are of interest. In particular, there exist preferred transition paths between regimes, as was shown by analyses of observational data (Kimoto and Ghil 1993b; Plaut and Vautard 1994; Crommelin 2004). Supported by strong numerical evidence, Crommelin (2003) suggested to explain this phenomenon, which was also found in his barotropic model, within the framework of the multiple equilibria theory, namely by remnants of so-called heteroclinic cycles between two fixed points embedded in the two regimes. A heteroclinic connection is a solution that connects one invariant set (e.g., a fixed point) with another. Two or more heteroclinic connections that form a closed loop are called a heteroclinic cycle. Heteroclinic connections and cycles are, however, generally not robust to perturbations of system parameters, but they show up as near-connections, or remnants, if exact connections are nearby in parameter space. With a variant of the model used by Charney and DeVore (1979), Crommelin et al. (2004) demonstrated a bifurcation mechanism leading to the emergence of a heteroclinic cycle between the stationary high-index state and the stationary low-index state of the model. Based on these results, Kondrashov et al. (2004) and Selten and Brantstator (2004) put forward the hypothesis that the remnants of heteroclinic connections are the reason for the existence of preferred regime transition routes in a quasigeostrophic three-level T21 model developed by Marshall and Molteni (1993).

The hypothesis that preferred transition paths in baroclinic models are determined by ruins of heteroclinic connections relies on the assumption of steady states being the cause of regimes not only in barotropic, but also in baroclinic models. Indeed, Selten and Brantstator (2004) suggested the numerical determination of steady states in a three-level model and the approximate computation of the disturbed heteroclinic connections by using the technique developed by Crommelin (2003). Unfortunately, the correspondence between steady states and regimes in baroclinic models has not yet been convincingly shown. Attempts in that direction have been made by Reinhold and Pierrehumbert (1982) with a highly simplified two-level channel model and by Achatz and Opsteegh (2003) with a model based on the equations of a complex atmospheric circulation model projected onto a few leading empirical orthogonal functions (EOFs), thus accounting for baroclinic dynamics. In both cases, pronounced regime behavior has been observed, but none of the fixed points found was situated within or nearby a regime domain. Even most of the fixed points were situated far outside of the corresponding model’s attractor. These findings leave open the question whether the results are due to the specifics of the models or rather show a generic property of baroclinic atmospheric dynamics.

An explanation for the emergence of regimes as an alternative to the multiple equilibria paradigm has been proposed by Itoh and Kimoto (1996, 1997, 1999), who used a two-level T15 model and a five-level T21 model. In some range of a certain control parameter (static stability or horizontal hyperdiffusion), deviating from the standard value, the models possess multiple coexisting attractors. If the parameter is changed toward the standard value, these attractors lose their stability one after another, owing to catastrophic bifurcations (Thompson and Stewart 1986), until only one attractor remains. Further variation of the parameter leads to an explosive bifurcation (Thompson and Stewart 1986) of the remaining attractor, which suddenly grows in size and extends over the phase space regions formerly occupied by the other attractors. The ruins of the former smaller attractors show up as regimes. The irregular wandering of a phase space trajectory from one attractor ruin to another is called “chaotic itinerancy.” Preferred transition paths are determined by the order in which the attractors have lost their stability. This scenario does not necessarily require the existence of steady states embedded within the regimes. However, Itoh and Kimoto did not compare the modeled regimes with observations. Therefore, the question whether a similar mechanism of the emergence of regimes can be found in baroclinic atmospheric models with realistic regime behavior remains open.

This paper aims at analyzing the origin of regime behavior in a baroclinic model that possesses regimes similar to observations. To this end, a model is required that is complex enough to allow for a reasonable reproduction of observed regimes, but sufficiently idealized to permit a detailed investigation of the dynamical structures underlying the regime behavior. A hemispheric version of the quasigeostrophic three-level model developed by Weisheimer et al. (2003) is used for this purpose, which, in its original version, was equipped with an idealized orographic and thermal forcing. Here the forcing is changed so that the mean wintertime circulation over the NH and its variability are simulated as realistic as possible. The orography is accounted for with the full model resolution. The thermal forcing is adjusted by a newly developed iterative procedure based on the comparison of the output of model test integrations with observational data (cf. Sempf et al. 2005). After convergence of the tuning, a model integration over 1000 years, corresponding to perpetual winter conditions, is evaluated with respect to climate, low-frequency variability, and regime behavior. Steady states of the model are determined by a suitable numerical procedure in order to check whether they can be associated with the model’s regimes. It will be shown that the result is negative. Therefore, another mechanism of regime behavior is looked for. To this end, the change of regime behavior with reduced friction between the model atmosphere and earth’s surface is studied, revealing hints on chaotic itinerancy.

In section 2 of this paper, the model used will be briefly described. The determination of the model’s thermal forcing by a new iterative procedure is explained in section 3. In section 4, the 1000-yr integration is analyzed. In section 5 it is explained how to compute steady solutions of the model, and the results obtained are discussed. Experiments with decreased surface friction are presented in section 6, and section 7 concludes the paper.

2. Model description

The quasigeostrophic three-level model, described in detail by Weisheimer et al. (2003) and briefly by Sempf et al. (2005), simulates the spatiotemporal evolution of the large-scale atmospheric circulation over the NH in three homogeneous vertical layers of equal mass under perpetual winter conditions. For simplicity, we refer to the lowest layer as the 833-hPa level, to the middle layer as the 500-hPa level, and to the upper layer as the 167-hPa level in the following. The lower two levels represent the lower and middle troposphere, while the upper level mimics the upper troposphere and the lower stratosphere. The hemispheric streamfunctions at these levels are represented spectrally at T21 resolution.

The simple physical parameterizations contained in the model consist of orographic and thermal forcing terms and dissipative processes. The large-scale topography of the NH, represented at T21 resolution (Fig. 1), acts as orographic forcing. The thermal forcing is established by fixing radiative equilibrium temperature fields at the auxiliary model pressure levels 333 and 667 hPa. Using these fields, the diabatic heating/cooling is parameterized such that the 167–500- and the 500–833-hPa layer thicknesses tend to relax toward the radiative equilibrium, with a time scale of 22.7 days.

Another parameterization, namely a modified Ekman friction mechanism, damps the streamfunction at 833 hPa not toward zero, but toward a predefined surface forcing function (Houtekamer 1991). This zonally symmetric function represents baroclinicity near the earth’s surface due to the meridional gradient of surface temperature. It helps to enforce westerlies in the lowest level, which would be too weak otherwise. A time scale of 1.7 days is used for this combined surface drag/forcing. The determination of the shape of thermal and surface forcing functions is described in section 3.

As a purely dissipative process, a horizontal scale-selective ∇6 hyperdiffusion is adopted. Its strength is chosen so that, for example, waves with the maximal total wavenumber N = 21 are damped with an e-folding time of 12.9 h.

The chosen dissipative time constants are a result of extensive experimentation. Our thermal relaxation and friction time scales are similar to those adopted in a global QG three-level T21 model by Marshall and Molteni (1993), corresponding to 25 and 1.5–3 days (spatially varying), respectively. Our diffusion time scale is considerably shorter compared to other studies; for example, Marshall and Molteni (1993) used 2 days for N = 21.

As a result of the QG approximation, the vertical temperature lapse rate has fixed values at the two auxiliary levels 333 and 667 hPa. These values are set to 3.0 and 6.5 K km−1, respectively.

3. Model forcing

While the T21 orography is fixed and can easily be incorporated into the model, the determination of the thermal and surface forcing is a nontrivial issue. In a model as simplified as ours, realistic forcing fields are not likely to produce the most realistic results achievable with the model. On the other hand, a forcing optimized to produce most realistic model results might have an “unphysical” shape. The trade-off between these extrema depends on the problem or question posed on the model, and suitable criteria for the choice of forcing have to be formulated. The criteria adopted here were initially motivated by the work of Sempf et al. (2005), where the impact of orographic forcing and of zonal asymmetries in extratropical diabatic heating on the structure of the Arctic Oscillation (AO) was studied. There, it was necessary to force stationary waves in a physically correct manner, as far as possible. Therefore the nonzonal components of the radiative equilibrium temperature fields were adjusted in a way that, on the time mean, realistic patterns of nonzonal extratropical diabatic heating were acting in the model, while the zonal components of the fields were tuned to produce a zonal climatology as realistic as possible, regardless of how realistic per se the zonal forcing was. In fact, the zonal forcing of our QG model becomes somewhat unrealistic, in particular in the Tropics, after such an adjustment. But a realistic zonal climatology ensures adequate westerly flow against orography and appropriate flow instability conditions. As the model forcing adjusted in this way generates reasonable model climatology and variability (see section 4), it is used in this study of regime behavior. It has to be noted, however, that the diabatic heating in this simplified model is not interactive with the circulation variability in the same way as it is in reality. For example, latent heat release associated with traveling cyclones or with tropical deep convection is accounted for only on the time mean, only implicitly, and in a highly simplified manner.

For our model we have designed an automated iterative procedure that adjusts the zonally symmetric surface forcing function and the zonal and wave components of the radiative equilibrium temperature fields at 333 and 667 hPa simultaneously. It works as follows: For a given forcing, a test run is performed, and the model’s time-mean zonal wind profiles for the three model levels are compared with the observed wintertime (DJF) zonal winds computed from National Centers of Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data (http://www.cdc.noaa.gov), averaged with mass weighting over three vertical layers of approximately equal mass. According to arising differences, moderate changes in the zonally symmetric portions of the forcing fields are computed subject to a rule briefly described in the appendix. Additionally, the model’s time-mean nonzonal diabatic heating at 333 and 667 hPa is diagnosed a posteriori and compared with the nonzonal heating fields derived from NCEP–NCAR data by Nigam et al. (2000) at 300 and 700 hPa. The latter fields have been attenuated near the equator and are shown in Fig. 2. Corrections of the nonzonal portions of the equilibrium temperature fields are derived from the differences between modeled and observed diabatic heating patterns. Some details are also given in the appendix. After the corrections of forcing, the model is run again, repeating the procedure, until a convergence criterion is met.

4. Model results

All model results presented in this section are based on a 1000-yr perpetual-winter integration. The climatology of the model is discussed first, while its low-frequency variability and regime behavior are addressed later on.

a. Model climatology

The time-mean zonal wind profiles taken from the model simulation are presented in Fig. 3, together with zonal winds taken from the reanalysis data. As is visible, the agreement between the modeled and the observed profiles is almost perfect, showing the effectiveness of the tuning procedure. The wind speed maximum situated at about 30°N in the middle and upper level corresponds to the subtropical jet. Additionally, there is a hint on a bump in the upper-level profile at high latitudes, which is due to the stratospheric polar vortex captured by the layer averaging of the reanalysis data.

To analyze geopotential height fields, the daily streamfunction output for each model level has been converted into a time series of geopotential height fields by means of the nonlinear balance equation (cf. Kurgansky 2002). The left column of Fig. 4 shows the time-mean geopotential height fields for the three levels. The most important observed extratropical features are reasonably reproduced by the model. At 833 hPa, the model shows ridges above the orography maxima and troughs in the lee of them, similar to observations (not shown, but cf. Sempf 2005). The Icelandic and Aleutian lows are visible; the latter, however, is considerably weaker than observed. The trough over Eastern Europe is somewhat exaggerated. The middle and upper level exhibit climatological troughs near the east coasts of Asia and North America, in good agreement with observations. There is, however, a noticeable trough over the middle Pacific, which is absent in the observations. Generally, the model shows limitations in the Tropics, which is a well-known consequence of the QG approximation.

The standard deviations of the geopotential height fields are displayed in the right column of Fig. 4. At 833 and 500 hPa, well-developed maxima of variability are visible over the northern Atlantic and Pacific (the storm track regions), having counterparts in observations, but being shifted westward. At the 167-hPa level, most variability is confined to the polar region, mainly caused by variations in the strength of the polar vortex (see below), again in reasonable agreement with observations.

b. Low-frequency variability

We analyze the model’s low-frequency variability by means of an EOF analysis of the daily geopotential height data of the lowest model level for the whole hemispheric domain using area weighting. The first and second EOF are shown in Fig. 5. The first EOF, explaining 14.5% of the total variance, strongly resembles the AO pattern (Thompson and Wallace 1998). The oceanic centers of action are, however, slightly shifted to the west compared to the observed AO pattern. The model is able to reproduce the AO and its extension into the upper troposphere/lower stratosphere with accuracy. At 500 hPa as well as at 167 hPa (not shown), the model’s first EOF possesses an annular structure similar to the 833-hPa pattern. The upper-level pattern represents a strengthening/weakening of the polar vortex. The first principal components (PCs) of the different levels are highly correlated (cf. Sempf et al. 2005; Sempf 2005).

The second EOF explains 7.0% of the variance and exhibits a dipole structure. A similar structure of the second EOF is found when analyzing observational daily SLP fields or geopotential height fields of, for example, the 850-hPa surface. However, the observed dipole centers are located farther to the east over the North Pacific and North Atlantic (not shown).

For the purpose of checking to which extent intrinsic decadal variations exist in the model, power spectra of the first and second PC of the lowest level geopotential height have been computed using an average over 16 windowed, overlapping periodograms. At every frequency, the uncertainty (standard deviation) of this spectral estimate can be assessed as being roughly 1/4 the value of the power density (Welch 1967). The spectra are shown in Fig. 6. The spectrum of the first PC, which is associated with the AO-like pattern, is red with maximum power on the interannual and decadal time scale. For the second PC, the spectrum is similar, but more white. These spectra are characteristic of fully developed chaotic model behavior and indicate the atmosphere by itself being capable of generating pronounced variability on ultralong time scales. This confirms the results of more idealized model studies (James and James 1989, 1992; Kurgansky et al. 1996; Dethloff et al. 1998; Weisheimer et al. 2003; Crommelin 2002).

c. Regime behavior

For regime detection we adopt the method described by Kimoto and Ghil (1993a) and Hsu and Zwiers (2001). It is based on (i) probability density estimation in a space spanned by two PCs and (ii) determination of the domains in that space that possess significantly higher probability than to be expected in the case of a bivariate red noise process with the same time-mean covariance matrix and autocorrelation at lag 1 day as the two PC time series. The significance is tested by means of 1000 Monte Carlo simulations. The significant areas are assumed to correspond to circulation regimes. For density estimation, we use an adaptive kernel estimator; the smoothing parameter is chosen by means of least squares cross validation (Silverman 1986). To save computational resources, the analysis has been restricted to the first 100 years of the model integration. Figure 7 shows the probability density estimate for the PC1–PC2 space of 833-hPa geopotential height, together with the significance contours. Two major regimes are found, denoted NAO and AO+. The composite anomaly patterns corresponding to the two regimes are shown in Fig. 8. The NAO regime pattern has a strong negative anomaly over the North Atlantic, similar to the negative phase of the North Atlantic Oscillation (NAO; Hurrel 1995). This anomaly, however, is shifted westward compared to the observed NAO pattern. The positive anomaly over Greenland is weaker than observed, and the maximum is shifted toward the Siberian coast. In addition to the NAO-like structure, there is a weak negative anomaly in the North Pacific region. The AO+ regime pattern generally bears resemblance to the positive phase of the AO, but with the oceanic centers shifted westward. Its amplitude is considerably weaker than that of the NAO pattern. Overall, the anomaly patterns have similarities with the regimes “blocked NAO” (BNAO) and “zonal NAO” (ZNAO) found by Kimoto and Ghil (1993a) when analyzing the observed daily 700-hPa height fields and, to some extent, with the clusters D and B found by Corti et al. (1999) in monthly mean 500-hPa height fields. Thus, the model is realistic enough to mimic observed phenomena, and its relative simplicity can be exploited to attempt explaining them.

5. Steady states

As a first attempt to give a dynamical explanation of the regimes found, we have followed the suggestion by Selten and Brantstator (2004) and computed steady solutions of the model equations in order to check whether they resemble the regime centroids. Steady states have been determined following the principle of a functional minimization method introduced by Brantstator and Opsteegh (1989) for the conservative barotropic vorticity equation and adapted by Crommelin (2003) for the forced–dissipative barotropic model. Every steady solution is equivalent to a zero of a nonnegative functional which, in the case of our three-level model, describes the squared tendency of the QG potential vorticity, summed up over the three model levels and integrated over the sphere, and reads
i1520-0469-64-3-887-e1
Here ψj (j = 1, 2, 3) denotes the streamfunction at the jth model level and j the time derivative of the QG potential vorticity at level j. Using the spectral representation, the functional (1) becomes a function of the spectral coefficients of the streamfunctions, and it is minimized using a numerical standard procedure for function minimizing, namely the routine E04UCF of the NAG Fortran Library (1999), a quasi-Newtonian method. For reasons of efficiency, this routine requires an analytical expression of the gradient of the function to be minimized. In analogy to the calculations shown by Brantstator and Opsteegh (1989) and Crommelin (2003), the gradient of the functional F can be derived by variational calculus (Sempf 2005).

Owing to the existence of numerous local nonzero minima of the functional (1), we have sampled a huge amount of different initial states for minimization from the model attractor and its close neighborhood. The first series of initial states consists of every fifth day of the first 100 yr of the model integration, 7200 states in total. The second series, having the same length, consists of every fifth day of the 10-day low-pass-filtered model trajectory. Furthermore, two series of 7200 randomly chosen states have been used. In the first random series, all 693 real variables describing the model state are normally distributed with the same mean and standard deviation as in the case of the model integration. The second random series has been generated in the same manner as the first, but using 1.5 times the standard deviation, in order to find potentially existing fixed points at the periphery of the attractor.

The result of this resource-demanding computation is sobering. A few percent of the initial states have converged to true steady solutions, but only six different steady states have been found, and all of them are very unrealistic and located far outside of the model’s attractor. Thus, they have no similarity with the regime centroids. With one exception, all steady states have a middle- and upper-level subtropical jet about twice as strong as in reality. Figure 9 shows the zonal wind profiles of a typical steady-state example with strong subtropical jet. An additional, weaker jet in the middle and upper level exists in the polar region, and at the lowermost level zonal winds are generally weak. All steady states are highly baroclinically unstable with disturbances growing on a time scale of one day or less.

It has to be noted that our results do not rule out with absolute certainty the existence of steady states in the close neighborhood of the attractor, or even of the regime centroids. But it appears very unlikely that such states did exist but could not be found by the functional minimization procedure although all the initial states were distributed on or nearby the attractor, whereas it was possible to find the remote fixed points. Therefore we are convinced that steady states in the close neighborhood of the attractor do not exist. For crosschecking, we have undertaken the same computation with the model version used by Sempf et al. (2005), which differs from the one used here only by a slightly stronger surface friction (with a time scale of 1.4 days) and, correspondingly, by a slightly different thermal forcing due to tuning. This experiment has confirmed our results, as 17 steady states have been found, all of them having the exaggerated subtropical jet.

The question arises why this strongly anomalous appearance of the steady states emerges, while steady states of barotropic models can correspond to fairly realistic circulations (Crommelin 2003). This question is addressed in a companion paper (Sempf et al. 2007), and here we only give a tentative explanation. In addition to the stationarity of the spatial momentum distribution, steadiness of a baroclinic circulation implies the time independence of the spatial distribution of air temperature. This means that the diabatic heating has to be compensated immediately and everywhere around the globe by the divergence of the instantaneous heat flux. This strongly constrains the possibilities for the existence of a steady state. Presumably, the existence of steady states that are far away from thermal equilibrium and look like realistic atmospheric circulations is very unlikely. Indeed, the discovered steady states with their strong subtropical jets appear to be rather close to thermal equilibrium, at least in the zonal mean. It is the time dependence of the atmospheric circulation that enables the atmosphere (acting as a heat engine) to remain far away from thermal equilibrium, namely due to the poleward heat transport by transient eddies.

6. Increased regime behavior by decreased surface friction

After rejecting the multiple equilibria picture for the three-level model, the chaotic itinerancy hypothesis put forward by Itoh and Kimoto (1996, 1997, 1999) remains to be tested. According to this hypothesis, regimes result from the unification of coexisting attractors when a certain control parameter is varied. Unfortunately, it has been found that it is impossible to reproduce the attractor merging scenario described by Itoh and Kimoto in our three-level model directly by simply varying the horizontal hyperdiffusion, or static stability, or any other parameter. No hints on a breaking apart of the single model attractor into multiple coexisting ones could be found when parameters were changed. Rather, the change of the parameters modified the model’s climate considerably. This climate change was accompanied by a distortion and blurring of the model’s circulation regimes. This negative result has led us to the assumption that the change of the model climate has to be compensated either by variation of additional parameters or by modification of the forcing in order to maintain the structure of the regimes and, possibly, to observe hints on the existence of an Itoh–Kimoto scenario in the three-level model. Restoring the model climate by updating the thermal forcing can readily be accomplished by repeating the iterative tuning procedure described in section 3 after every change of the control parameter. In the following, we report on a series of such kind of experiments where the surface friction parameter is decreased and hints on the existence of an attractor merging scenario can be observed.

Four experiments, called A, B, C, and D, have been performed. In each experiment, a surface friction time scale longer than the original time scale of 1.7 days adopted in the control simulation has been used, and the tuning procedure has been repeated. Afterward, the model has been integrated over 1000 years for each experiment. In experiment A, the time scale is 2.4 days, it is 2.8 days in experiment B, 3.0 days in C, and 3.3 days in D.

While the time-mean circulation is approximately equal in all four simulations, the variability considerably increases, particularly in the polar region, when the surface friction is decreased (figures not shown). Figure 10 shows the first EOF of 833-hPa geopotential height for the four experiments. In experiment A, the similarity of the pattern with that from the control simulation (left panel of Fig. 5) is greatest. When the surface friction is decreased further, the zonal symmetry of the patterns slightly increases, while their amplitude becomes much greater. All patterns still have some similarities with the structure of the AO. It can be concluded that the decrease of frictional dissipation leads to a dramatic increase of the AO-like variability. Additionally, the AO becomes more dominant. In experiment A, the first EOF explains 16.0% of the total variance, more than in the case of the control simulation. The variance fraction further increases to 18.8% in experiment B, to 26.4% in C, and even to 35.3% in D. The increase of variability in the polar region is mostly explained by the strengthening of the AO.

All second EOF patterns of 833-hPa geopotential height for the four experiments (not shown) have a wavenumber-1 structure similar to the corresponding pattern from the control simulation (right panel of Fig. 5). The amplitude of the patterns also increases with decreasing friction, but not as drastically as in the case of the first EOF.

The respective power spectra of the first PC of 833-hPa geopotential height reveal a strong increase of decadal variability with decreasing friction (not shown). This is in line with the visual impression given by 5000-day segments of the corresponding time series for the four experiments, as shown in Fig. 11. From experiment A to D, the time series gain more and more low-frequency content. Furthermore, regime behavior becomes more and more pronounced and is most prominent in experiment D, where the model can reside for hundreds of days in either the positive AO or the negative AO regime, followed by a rather abrupt transition to the respective other regime.

The enhancement of regime behavior when decreasing friction also becomes manifest in the probability densities, which have been estimated in the respective space spanned by the first two PCs (shown in Fig. 12 together with the significance contours). Due to the similarities of the corresponding EOFs for all experiments, all these subspaces are comparable. The distribution from experiment A is slightly skewed toward positive values of the first PC, similar to the distribution from the control simulation (Fig. 7). This skewness becomes stronger with decreasing friction. Out of the center of the distribution, a shoulder is growing toward negative values of the first PC, weakly visible in B, but already clearly seen in C. At last, in D, the shoulder has turned into a second local maximum—the distribution has become bimodal.

In experiment A, a third regime in the upper-left corner of the corresponding density plot appears, which is absent in the density from the control simulation (Fig. 7). In experiment B, the two regimes with negative PCI values (negative AO phase) are about to merge. This merging is completed in experiment C. Here, even a connection with the positive AO regime is established so that there is only one connected domain of unexpectedly high probability. On the other hand, this connection is situated in an area with very low PC2 values and generally low probability, so experiment C can well be interpreted as having two regimes. The “bridge” between the two regimes might be a hint on a remote transition path. In experiment D, a distinct two-regime structure is visible. From all results of the experiments, we conclude that decreasing the surface friction leads to an increase of (i) the AO amplitude, (ii) the regime persistence, and (iii) the regime separation.

These findings, in particular (ii) and (iii), are an indicator, albeit not a strict proof, of an attractor merging scenario. Intuitively one might expect the system to break into two dynamically disconnected parts, or coexisting attractors, if the surface friction is decreased further. Unfortunately, we could not investigate this case owing to a dramatically increasing sensitivity of the model with respect to forcing changes and, consequently, a failure of the tuning procedure. So we have to rely on theory now. In principle, disruption of a single chaotic attractor into two parts, or equally, the opposite process of merging of two coexisting chaotic attractors to form a single one, can occur by a single bifurcation, namely the attractor merging crisis (Grebogi et al. 1982, 1983). In this bifurcation, the two coexisting attractors simultaneously collide with the boundary between the two basins of attraction,1 and a single, bigger attractor emerges, extending over the phase space regions occupied by the two former attractors. But this simultaneous collision, and so the attractor merging crisis, is not a generic case and can only occur in systems with symmetries. In general cases, one attractor will always be the first to collide with the basin boundary when the control parameter is varied. After such a boundary crisis (Grebogi et al. 1982, 1983), the attractor has lost its stability, as the basin has become leaky, so that trajectories can escape after a (sometimes very long) transient period, and are ultimately captured by the remaining attractor. After further variation of the control parameter, if the remaining attractor collides with the set to which the former basin boundary has evolved to, it undergoes an interior crisis (Grebogi et al. 1982, 1983). As a result, it suddenly grows in size and annexes the region formerly occupied by the other attractor. The irregular transitions between the old and the new attractor part are referred to as crisis-induced intermittency, or, more generally, chaotic itinerancy. The effect of the two subsequent bifurcations is the merging of the two attractors and the establishing of a pronounced two-regime structure. Right after the interior crisis, regime transitions occur rather seldomly, but become more frequent when moving away from the bifurcation point. Indeed, the characteristic transient lifetime of the old regime diverges at the bifurcation point and decays in its neighborhood, often according to a power law (Grebogi et al. 1987; Ott 1993), or with the square root of an exponential function (Grebogi et al. 1985). We hypothesize that experiment D is in a condition rather close to an (inverse) interior crisis, and note that the decrease of regime persistence with increasing surface friction is in qualitative agreement with the behavior to be expected when approaching the critical parameter value.

7. Summary and discussion

A quasigeostrophic, hemispheric three-level model using the T21 orography of the NH has been adapted to NCEP–NCAR reanalysis data using an automated iterative procedure. The zonal part of thermal forcing has been adjusted to produce a realistic wintertime zonal wind structure, while the nonzonal part of thermal forcing has been tuned in such a way that, on the time mean, the nonzonal extratropical diabatic heating acting in the model is equal to wintertime observations. In a 1000-yr perpetual winter simulation, the model has shown to reproduce the large-scale extratropical wintertime circulation with adequate accuracy. It possesses reasonable climatology, including time-mean and standard deviation patterns. Furthermore, it is able to reproduce the AO signature in the troposphere and the upward extension of the AO. The simulated AO is associated with pronounced interannual and decadal variability. Decadal variations have also been detected in the observed AO index, for example, by Tanaka (2003). Of course, the observed fluctuations are not only due to internal atmospheric variability, but also caused by interactions between the atmosphere and other climate subsystems, and by changes of external, for example, anthropogenic, forcing on the whole climate system. Yet the model results suggest that natural variations due to internal nonlinear dynamical processes within the atmosphere might contribute to observed decadal climate variability to a substantial part.

The model is shown to possess two circulation regimes. One of them resembles the negative phase of the NAO, the other one is similar to the positive of the AO, in reasonable agreement with regime patterns found in observational studies (Kimoto and Ghil 1993a; Corti et al. 1999). We have tested whether these regimes can be explained by the presence of steady states, which have been searched for by means of a functional minimization procedure. The result is negative, but it is in qualitative agreement with the findings by Reinhold and Pierrehumbert (1982) and Achatz and Opsteegh (2003). All steady states found are located far away from the model’s attractor, although every search attempt has started in the close neighborhood of it. Therefore, there is no relationship between the steady states and the regime centroids. Most of the steady states have a strong subtropical jet and are seemingly close to thermal equilibrium. We supposedly have found a general property of baroclinic models. Indeed, in contrast to barotropic models, the additional condition of instantaneous heat balance has to be fulfilled for a model state to be steady, and this constraint might be not very likely to allow for a steady flow that strongly deviates from thermal equilibrium, as realistic circulation states always do. It seems that the multiple equilibria paradigm is not applicable to baroclinic dynamics. Consequently, it is not possible to explain preferred regime transition routes in baroclinic models by disturbed heteroclinic connections between steady states, as suggested by Kondrashov et al. (2004) and Selten and Brantstator (2004), after Crommelin (2003) found evidence of such connections in a barotropic model. However, our results do not rule out the existence of heteroclinic connections between other simple invariant sets, such as periodic orbits or tori.

The steady-state analysis has shown that the regime behavior of the baroclinic three-level model does not originate from the existence of steady states, but must be a result of an inhomogenous large-scale geometry of the attractor. Hints on the origin of this geometry, in turn, are given by a series of four experiments with successive reduction of the surface friction parameter and retuning the model. In these experiments, a strong increase of AO-like variability has been detected when the surface friction is decreased. But, above all, the regime persistence and separation drastically increases, visible through inspection of the PC1 time series of 833-hPa geopotential height and through an increasing bimodality of the probability density in the PC1–PC2 space. Therewith, the decadal fraction of the variability increases. We have argued that such behavior is characteristic of approaching an (inverse) interior crisis, which we assume to be a part of a bifurcation scenario leading to a merging of at least two coexisting attractors. Such a scenario is not entirely new, as Itoh and Kimoto (1996, 1997, 1999) have reported on similar scenarios occurring in a two-level T15 model and a five-level T21 model. What is new here is the fact that hints on the scenario have been found in a model that possesses regimes resembling well-known atmospheric circulation patterns, such as AO and NAO. Furthermore, we hypothesize that such a scenario is common among all fairly realistic atmospheric circulation models, provided that a suitable control parameter is chosen. Possibly, this parameter needs to be constructed in such a way that it influences other parameters or forcing terms in a complicated fashion in order not to dramatically affect the model climate and the spatial structures of the regimes as the parameter is varied. The hypothesized bifurcation scenario is schematically depicted in Fig. 13 for the case of two regimes, but can readily be generalized for the case of more regimes. For a certain value of the control parameter assumed to be lower than the default value, the system is intransitive because it possesses two coexisting attractors X and Y, which, for simplicity, are assumed to be chaotic. As the parameter is raised and exceeds a critical value, attractor Y becomes unstable by a boundary crisis; hence it is no longer an attractor. However, long-lived chaotic transients may occur in the phase space region formerly occupied by attractor Y. As the control parameter is further increased, it eventually passes a second critical value, where an interior crisis of the remaining attractor X occurs. The latter suddenly grows in size and incorporates the phase space region that formerly belonged to attractor Y. As a result, a merged attractor with a pronounced two-regime structure emerges. The phenomenon of irregular transition between the ruins of formerly coexisting attractors is referred to as crisis-induced intermittency or chaotic itinerancy by the dynamical systems community. Right after crossing the second critical parameter value, the connection between the two regimes is weak, and transitions occur rather seldomly. Systems with this property have been called almost intransitive by Lorenz (1968, 1976). But when the parameter is further increased toward the default value, the attractor undergoes a smooth deformation by which the connection between the regimes becomes stronger and transitions become more frequent, in accordance with theory (Grebogi et al. 1985, 1987; Ott 1993). At the default setting, the almost-intransitive property has weakened; that is, the regime behavior is less obvious, but can be detected by suitable statistical methods. Still, the attractor bears the traces of the bifurcation scenario. The results from experiments A through D may be explained by a backward deformation from weakened to strong almost intransitivity, or from the uppermost part of Fig. 13 to the next part below, as the regimes become increasingly more persistent and more clearly separated than in the control simulation. Furthermore, we note that the two assumed subsequent bifurcations, which lead to the merging of two coexisting attractors, can be observed in a barotropic model as well as during a smooth transition from barotropic to the three-level baroclinic flow, as reported by Sempf et al. (2007).

As pointed out by a reviewer, our observation that the regime behavior becomes clearer when a dissipative parameter is decreased is somewhat unexpected a priori. Indeed, decreasing the dissipation should lead to an increase in the attractor’s size and also in its dimension and thus give more homogeneous phase space structures, whereas one expects clustering in phase space when dissipation is strong. This paradox is resolved by the fact that the growth of the attractor’s size (namely, the increase of variability), which we have indeed observed when reducing surface friction, is highly anisotropic in our case. Considering our attractor to consist of two connected parts, we state that, during friction decrease, (i) the distance between the centroids of the two parts grows, as indicated by the increase of the absolute AO amplitude, and (ii) the two parts themselves grow, as shown by the increase of the absolute amplitude of higher EOFs. But the growth along the attractor’s principal axis is stronger than in other directions, as demonstrated by the increase of the variance fraction explained by the AO (cf. section 6). That means that the aspect ratio of the attractor flattens and the two parts must eventually disrupt. In addition, we can observe similar results in our model when a certain dissipative parameter is raised. We performed experiments like A–D but with introducing and then gradually increasing a vertical turbulent friction in the interfaces between the model levels (an internal friction in the model), instead of decreasing the surface friction. Thereby, we observed a similar increase of regime separation, with the only important difference being that the amount of variability remained approximately constant in these additional experiments. These results should support the generality of the conclusions drawn from our surface friction experiments.

Acknowledgments

We cordially thank E. DeWeaver and S. Nigam for sending us the diabatic heating data, F. Molteni for kindly providing us the orographic dataset, A. Weisheimer for support in the model handling and many useful hints, and two reviewers for their helpful comments and suggestions.

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APPENDIX

Forcing Function Adjustment

All forcing manipulations are done on the NH domain of a Gaussian grid. Afterward, an antisymmetric reflection onto the Southern Hemisphere domain is performed, followed by transform into spectral space. A smoothness constraint at the equator is imposed.

When correcting the zonal parts of the forcing functions, modifications of the surface forcing are done at first. At every grid point, the meridional gradient of the surface forcing is changed by an amount proportional to the difference between the observed zonal wind in the lowermost atmospheric layer (and at the corresponding latitude) and the model’s zonal wind at the lowest level and the same latitude. As a result, stronger zonal winds in the lowest level will be forced in the next model run where they have been too weak, and vice versa.

This adjustment of surface forcing acts barotropically on all model levels to a first approximation, but the effect is desired only at the lowest level. Therefore all corrections added to the surface forcing are subtracted from the zonal 667-hPa forcing. Afterward, this forcing is adjusted in the same way as the surface forcing, but according to differences between observed and modeled middle-level zonal winds. These corrections also affect the upper level and are subtracted from the zonal 333-hPa forcing, which is calibrated at last, using observed and modeled upper level zonal winds.

For the adjustment of the nonzonal forcing, the time mean of the nonzonal part of diabatic heating at 333 and 667 hPa acting in the model during the test run is computed. To each of the two equilibrium temperature fields, a correction proportional to the difference between the modeled and observed nonzonal heating at the corresponding level is added.

Fig. 1.
Fig. 1.

T21 orography of the NH. Contour interval is 500 m; zero contour omitted.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 2.
Fig. 2.

Wintertime (DJF) extratropical nonzonal diabatic heating at (left) 300 and (right) 700 hPa derived from NCEP–NCAR reanalysis data and used in the QG three-level model. See text for details.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 3.
Fig. 3.

Modeled and observed time-mean zonal wind profiles for the three model levels and corresponding vertical layers, respectively.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 4.
Fig. 4.

(left) Mean geopotential height of the (top) 167-, (middle) 500-, and (bottom) 833-hPa model level. (right) Geopotential height standard deviation of the (top) 167-, (middle) 500-, and (bottom) 833-hPa model level.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 5.
Fig. 5.

(left) First and (right) second EOF of 833-hPa geopotential height.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 6.
Fig. 6.

Power density spectrum of the (left) first and (right) second PC of the 833-hPa geopotential height.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 7.
Fig. 7.

Estimated probability density in the space spanned by the first two PCs of the 833-hPa geopotential height. Contour interval is 0.01; zero contour omitted. The thick lines enclose areas with significantly higher probability than to be expected in the case of the adapted red noise process (at the 95% level), and thus mark the circulation regimes. The regimes are denoted as indicated.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 8.
Fig. 8.

Composite anomaly pattern of the (left) NAO regime and the (right) AO+ regime at the 833-hPa level.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 9.
Fig. 9.

Zonal wind profiles of a typical steady-state example.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 10.
Fig. 10.

First EOF of the 833-hPa geopotential height for experiments (upper left) A, (upper right) B, (lower left) C,and (lower right) D. For a better comparison of pattern structures, individual contour intervals are used for each pattern.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 11.
Fig. 11.

The first 5000 days of the first PC of 833-hPa geopotential height for experiments (upper left) A, (upper right) B, (lower left) C, and (lower right) D.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 12.
Fig. 12.

Estimated probability density in the space spanned by the first two PCs of 833-hPa geopotential height for (upper left) experiment A, (upper right) experiment B, (lower left) experiment C, and (lower right) experiment D. Contour interval is 0.01; zero contour omitted. The thick lines enclose areas with unexpectedly high probability, as in Fig. 7.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

Fig. 13.
Fig. 13.

The bifurcation route to regime behavior hypothesized to be common among realistic atmospheric models.

Citation: Journal of the Atmospheric Sciences 64, 3; 10.1175/JAS3862.1

1

A basin of attraction is the set of all points in phase space that are attracted by the attractor to which the basin belongs.

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