## 1. Introduction

The master equation is a prognostic equation for the probability density function (PDF). In this paper, a discrete time approximation of the master equation is used in discretized phase spaces spanned by climate variables. Its coefficients give the probability for transitions between states. In an empirical master equation (EME), the coefficients are estimated from time series of the variables. This technique is not new in the atmospheric sciences and has successfully been applied, for example, to the issue of weather regime transitions (Spekat et al. 1983; Crommelin 2004), to ENSO (Fraedrich 1988; Pasmanter and Timmermann 2002), and to the equatorial components of the global angular momentum of the atmosphere (Egger 2001). The PDF forecasts given by the EME can be used for making probabilistic predictions. Moreover, the evolution of the PDF may provide insight into the underlying processes. The latter utility is used in this part of the paper to gain insight into stratospheric dynamics.

The stratosphere plays a crucial role in the climate system through its radiative and dynamical processes. The stratosphere undergoes a large intraseasonal and interannual climate variability such as the internally driven quasi-biennial oscillation (QBO) of equatorial zonal wind in the stratosphere [Reed et al. (1961); Veryard and Ebdon (1961); see the review by Baldwin et al. (2001)] the decadal changes attributed to the 11-yr solar cycle (SC) (see Labitzke and van Loon 1999), and the impact of volcanic eruptions (e.g., Labitzke and van Loon 1999). A major aim of climate research is the search for atmospheric modes of oscillation with long time scales. The QBO of stratospheric zonal wind above the equator is characterized by alternating and downward propagating phases of westerlies and easterlies with a period generally between 22 and 34 months. The QBO represents the most striking example of a long-term periodic atmospheric oscillation besides the orbitally forced annual components with their harmonics. The period of the SC varies between 9.5 and 12 yr (Labitzke and van Loon 1999). Its signal is strongest above the Tropics in the summer hemisphere and the QBO is believed to modulate the global signal during Northern Hemisphere winter (Labitzke 2001). The mechanisms behind the relatively strong response of stratospheric climate to small variations in solar irradiance are not yet fully understood nor is the mechanism of interaction of the QBO and the SC (e.g., Labitzke and van Loon 1999; Salby and Callaghan 2000; Rind 2002; Gray et al. 2004). In recent years there has been a growing realization that stratospheric effects can influence tropospheric climate (see Houghton et al. 2001, 432–435). Moreover, there is increasing evidence that the inclusion of stratospheric processes may improve the skill of long-range tropospheric weather prediction (Baldwin and Dunkerton 2001; Baldwin et al. 2003). For example, the QBO influences the stratospheric circulation (e.g., Holton and Tan 1980), the atmospheric ozone that affects the intensity of the ultraviolet light reaching the biosphere (e.g., Stolarski et al. 1991), and its signal can be found even at the NH extratropical surface (Coughlin and Tung 2001). The occurrence and timing of stratospheric sudden warmings are also influenced by the QBO and the SC (Gray et al. 2004) and pronounced changes in the stratospheric north polar vortex are often followed by changes in surface weather (Thompson and Wallace 2001; Thompson et al. 2002). The dynamical coupling of the troposphere with the stratosphere^{1} in the NH is captured by the northern annular mode (NAM; see Thompson and Wallace 1998; Baldwin and Dunkerton 1999; Thompson and Wallace 2000). In the winter stratosphere the annular mode is linked to the temperature and strength of the polar vortex. In the lowermost troposphere, the NAM appears as the Arctic Oscillation and, in particular, over the North Atlantic as the North Atlantic Oscillation [NAO: see also Wallace (2000)], whose phases are connected to weather regimes and precipitation patterns in NH midlatitudes (e.g., Qian et al. 2000; Thompson et al. 2002). A number of studies report evidence that winter anomalies of the stratospheric circulation propagate downward into the troposphere as anomalies of the annular modes (e.g., Baldwin and Dunkerton 1999, 2001). However, there is still controversy with regards to the relative role of stratosphere and troposphere (e.g., Haynes et al. 1991, 1996; Egger 1996; Black 2002; Ambaum and Hoskins 2002; Haynes 2005). We use EMEs to gain insight into the underlying processes of stratospheric dynamics.

The numerical properties of the EME have been studied in the first part of this paper (Dall’Amico and Egger 2007, hereafter Part I). In Part I, we have studied how the quality of the EME depends on the grid size used to discretize the phase space, on the length of the time series, on the time resolution, and on the number of variables used, that is, the dimensionality, and the selection of the variables. The number of variables, Λ, should be kept small to allow for a reasonably fine grid size. Three-dimensional EMEs are currently feasible. In Part I, we found that

- the grid size,
*Dq*, should be initially chosen in order to partition the phase space into a few hundred cells and then adjusted according to the quality of the resulting EME; - for a stationary stochastic system, there is a threshold length of the time series beyond which the quality of the EME does not improve;
- an increase of the time step,
*Dt*, leads to a reduction of the numerical diffusion; - the projection onto a lower-dimensional phase space calls for caution when the EME is not able to capture the behavior of the studied system in an acceptable way. If feasible, the dimension of the EME should be increased.

In this part of the paper, the EME is implemented within the stratospheric context as described in section 2. Section 2 builds on section 2 of Part I, and several quantities introduced there are used in this part of the paper.^{2} The methodology described in section 2 can be applied to any issue in the atmospheric sciences. In sections 3a and 3b, the EME is applied for studying the influence of the QBO and the SC on the temperature of the Arctic middle stratosphere; the relationship between the NAM in the stratosphere and in the troposphere is addressed in section 3c. The conclusions are outlined in section 4. This two-part paper is based on work presented in Dall’Amico (2005).

## 2. Stratospheric EMEs

*q*

_{1}, . . . ,

*q*, . . . ,

_{λ}*q*

_{Λ}, is discretized into equal-sized cells with grid size

*Dq*. The discrete time master equationis used to predict the probability density

*f*(

_{i}*t*

_{n}_{+1}) in cell (

**i**) = (

*i*

_{1}, . . . ,

*i*

_{Λ}) at time

*t*

_{n}_{+1}=

*t*

_{0}+ (

*n*+ 1)

*Dt*, where

*Dt*is the time step. The transition coefficient

*W*

^{i}

_{i′}gives the probability of a transition from cell (

**i**) to (

**i**′) (see Zwanzig 2001, 61–63) within

*Dt*. Though linear in

**f**, Eq. (1) may capture nonlinear relationships between the variables. In an EME, the transition coefficients are estimated as the relative frequencies of transitions observed in a time series of the variables as discussed in section 2 of Part I.

The amount of data available in observational records is in most cases insufficient to introduce time-dependent (e.g., monthly varying) transition coefficients. Hence the time series, assumed to stem from ergodic processes, are normalized with respect to their standard deviation and deseasonalized by subtracting the mean and the yearly and half-yearly harmonics. This standardization of every variable allows for cubic cells.

In section 2 of Part I, we defined *R _{w}* as the weighted average ratio between the half-width of the confidence intervals of the transition coefficient estimates and the estimates themselves. For larger (smaller) grid sizes, the cells will include more (less) observations, leading to lower (higher) values of the ratio

*R*. Values of this ratio below 0.4 can be considered acceptable; values up to 0.6 should lead either to much caution in the study or to the use of a coarser grid size; the latter applies to higher values. The computer resources limit the choice of a fine grid size, whereas the significance problem limits both the grid size and, less restrictively, the time resolution. In Part I we found that numerical diffusion increases as the time step is reduced. To use a coarse time step for the EME without losing part of the information inherent in a highly resolved time series through time-averaging or filtering,

_{w}*Dt*is chosen as a multiple, ϒ, of the time resolution of the time series,

*Dt*

_{TS}. A set of ϒ time series is developed, with the

*υ*th time series starting with the

*υ*th observation and (ϒ − 1) values being skipped before writing down the following value. Cell transitions are counted in these ϒ time series on the basis of the ϒth value observed after the current value. A stationary estimate of the transition coefficients may not be achieved for very large ϒ values.

An acceptable *R _{w}* value is not a guarantee that the EME correctly captures the dynamics of the studied system as seen in the data. To test this, correlation functions as delivered by the EME (see section 2 of Part I) are compared to those obtained from the data (e.g., Cencini et al. 1999; Egger 2001). A disappointing mismatch may occur if a higher-dimensional system is studied, suggesting, if feasible, the introduction of a further variable.

**q̂**= (

*q̂*

_{1}, . . . ,

*q̂*, . . . ,

_{λ}*q̂*

_{Λ}):where

*q̃*is the

_{λ}*λ*coordinate of the center of the cell in question, and the standard deviationassociated with the PDF. The evolution of the mean position describes a trajectory in phase space. Insight in the relationship between the variables can be gained by studying trajectories obtained from EME runs that begin with unit probability in cells representing various states of the system.

^{3}

In this part of the paper, EMEs are derived from time series of (up to three) climate variables. These variables are defined on the basis of spatial averages of meteorological quantities and of the difference of their values between selected regions. Time series of stratospheric climate variables for 1957–2002 are obtained from the 40-yr European Centre for Medium-Range Weather Forecasts Re-Analysis (ERA-40; Uppala et al. 2005, the data are available at www.ecmwf.int/research/era), examined at standard pressure levels and on a regular 7.5° × 7.5° latitude/longitude grid. Stratospheric dynamics are studied here with particular emphasis on the QBO, the SC and the NAM. Moreover, the Arctic stratosphere is of interest since its temperature is affected by these oscillations (e.g., Labitzke 1987). To study the effect of the SC, a time series of the solar radio flux at 10.7 cm is considered; this flux is highly and positively correlated with the extreme ultraviolet radiation varying with the SC, which affects the stratosphere (Labitzke and van Loon 1999), but can be measured from the earth’s surface (the time series used in this paper has been produced by the Dominion Radio Astrophysical Observatory of the Herberg Institute of Astrophysics, Canada, and is available at http://www.drao-ofr.hia-iha.nrc-cnrc.gc.ca/icarus/www/sol_home. shtml). As daily measurements of the solar radio flux are available, daily means for the climate variables are obtained by averaging the 6-hourly synoptic values of the reanalysis. Altogether, there is a data point per day for 16 436 days.

Two variables are needed for reproducing an oscillation like the QBO. If one estimates the transition coefficients in a one-dimensional case, transitions from any interval (but those at the extremes) are counted toward either phase; this leads to mere PDF diffusion in phase space. A first variable for the QBO, *Q*_{30}, is defined as the normalized and deseasonalized zonally averaged zonal wind at 30 hPa above the equator. We will refer to *Q*_{30} by mentioning “QBO index” without further specifications. The second variable, *Q*_{20}, is derived from the zonally averaged zonal wind at 20 hPa above the equator. These levels are close to the height where the QBO signal is greatest, that is, about 25 km (Baldwin et al. 2001, p. 218). (For any oscillation, the second variable may also be chosen as a time-lagged version of the first.) The variable for solar activity, *S*_{10.7}, is obtained from the solar radio flux at 10.7 cm. The temperature anomaly at 10 hPa, averaged for latitudes *ϕ* ≥ 75°N, is used to define the variable for the temperature anomaly in the Arctic stratosphere, *T*′. The correlation coefficient for *T*′ and the zonally averaged zonal wind at 55°N at 10 hPa is *r* = −0.6. Finally, variables for the NAM at 10 hPa (*A*_{10}, middle stratosphere), 100 hPa (*A*_{100}, lowermost stratosphere), and 850 hPa (*A*_{850}, lower troposphere) are obtained from the zonally averaged geopotential height difference between 45° and 67.5°N, that is, the geostrophic zonal wind in this latitude channel. A similar approach was first used by Lorenz (1951), who defined *U*_{55} as the zonally averaged sea level pressure difference between 45° and 65°N. Thompson and Wallace (2000) defined the NAM as the leading mode of month-to-month variability of hemispheric (from 20°N) geopotential height. The correlation between *U*_{55} and the principal component of the first EOF of ground pressure in NH is very high (*r* > 0.9: see Wallace 2000).

Many studies focus on the winter season when there is a strong dynamical coupling between the NH stratosphere and troposphere (Holton and Tan 1980; Labitzke 1987; Baldwin and Dunkerton 1999, 2001; Thompson et al. 2002; Ambaum and Hoskins 2002). However, winter EMEs turn out to be hard to interpret.^{4} Moreover, Labitzke (2004) suggests the introduction of the QBO throughout the full course of the year. In this part of the paper, deseasonalized time series for the complete period covered by ERA-40 (September 1957 to August 2002) are considered, where each season is equally represented. In the time series of *T*′, winter and spring anomalies dominate (not shown, see, e.g., Fig. 4.3 of Dall’Amico 2005).

## 3. Results

The effect of the QBO and that of the SC on the Arctic stratosphere are investigated first and then the relationship between the NAM at stratospheric and tropospheric levels is examined.

### a. QBO and Arctic stratosphere

The influence of the QBO on the temperature in the Arctic stratosphere is investigated with a three-dimensional EME for the variables: 30-hPa QBO (*Q*_{30}), Arctic stratospheric temperature (*T*′), and 20-hPa QBO (*Q*_{20}). Figure 1a shows a projection on the (*Q*_{30}, *T*′) phase plane of the distribution of observed states (the time series of these variables are shown in Fig. 4.2 and 4.3 of Dall’Amico 2005). Although most states have rather small temperature anomalies, there are large positive excursions linked to sudden warmings. During the west phase of the QBO, the QBO index, *Q*_{30}, is greater than about one standard deviation. A projection on the (*Q*_{30}, *Q*_{20}) phase plane is shown in Fig. 1d. The elliptical circumference form of the “data cloud” in Fig. 1d reflects the delay of the QBO phase at 30 hPa with regards to 20 hPa. As for the choice of the grid size, in Part I we suggested to begin with a grid size so fine as to partition the data cloud into a few hundred cells. A grid size *Dq* = 0.50 standard deviations leads to a partition of the data cloud seen in Fig. 1a into 235 cells, whereas with *Dq* = 1.00 one obtains just 61 cells. With a grid size *Dq* = 0.50 and a time step *Dt* = 30 days (i.e., ϒ = 30) the ratio *R _{w}* = 0.32; with a longer time step

*Dt*= 90 days one still obtains an acceptable ratio

*R*= 0.36. Lack of fine structures, the relatively low value of

_{w}*R*, and reasons of computer economy support the choice

_{w}^{5}

*Dq*= 0.50. After discretizing the phase space with a grid size

*Dq*= 0.50, isolines of the corresponding observed state density (denoted with

*ρ*in section 2 of Part I and defined with unit integral as the PDF), integrated along the perpendicular axis, are shown in Figs. 1b and 1e. The long-term, stationary PDF prediction from an EME with

*Dt*= 30 for an arbitrary initial condition is shown in Figs. 1c and 1f. The projection on both phase planes of the mean trajectory for this EME run is shown as well. Figures 1c and 1f are almost identical to Figs. 1b and 1e, respectively. In Figs. 1b and 1c, the more conspicuous temperature excursions are too rare to have much impact on the observed state density and the PDF. In Figs. 1b and 1c, the local maximum for QBO East (West) has a slightly positive (negative) temperature anomaly. This is in qualitative agreement with the relationship suggested by Holton and Tan (1980), who analyzed differences in composites of 50-hPa geopotential height and geostrophic wind in the winter months. Holton and Tan used monthly mean NH data for the period 1962–77 (16 yr) and found that the zonal mean geopotential height at high latitudes was significantly lower during QBO West than during QBO East (see also Baldwin et al. 2001, 218–220). If the grid size is decreased to

*Dq*= 0.20, however, the positions of the maxima in Figs. 1b,c shift toward

*T*′ = 0 (not shown). The mean

*T*′ for a given

*Q*

_{30}interval is given by the dashed–dotted line in Fig. 1b. The inclination of the dashed–dotted line between QBO East and QBO West is rather small and not uniform, suggesting a small difference between the two phases of the QBO. The mean trajectory in Fig. 1c loses rapidly its temperature anomaly and develops first toward the QBO West phase and then toward QBO East; in oscillating between the two phases, it displays a slight inclination, which is more uniform than that of the dashed–dotted line in Fig. 1b. The projection of the trajectory in Fig. 1f has a cyclonic evolution and, as expected, a contraction toward the climatological mean (i.e., the origin of the phase space); this motion reflects the necessity of using two variables for describing an oscillation. The trajectory bends toward lower

*Q*

_{20}values after the second time step, hinting at a sudden approach of the easterly shear after a smooth phase with westerly shear. This recalls the spike in Fig. 1d for positive values of both

*Q*

_{30}and

*Q*

_{20}.

Correlation functions for (*Q*_{30}, *T*′, *Q*_{20}) are shown in Fig. 2 as observed (solid lines) and as delivered by the EME (dashed for *Dt* = 30 and dash–dotted for *Dt* = 90). In the *r*_{Q30Q30} panel, the first minimum occurs at a lag of 13.6 months and the next maximum at a lag of 28.4 months (28.8 months at 20 hPa, not shown). The *r*_{Q30Q20} panel shows that *Q*_{30} and *Q*_{20} are highly correlated and that *Q*_{20} leads *Q*_{30} by about 11 weeks. The Holton and Tan relationship is reproduced qualitatively in the *r*_{T′Q30} panel, where the cross correlation is negative for lag zero,^{6} has a negative minimum for *Q*_{30} leading *T*′ by about 5 weeks, and displays a wavelike behavior with a period corresponding to that of the QBO. Altogether, this is a case where one variable is weakly correlated with the others and has a short time scale, whereas the other two variables are closely linked with a rather long time scale. The autocorrelation of *Q*_{30} (top left panel of Fig. 2) delivered by the EMEs decays faster for the finer time step *Dt* = 30 (dashed line) than for the coarser time step *Dt* = 90 (dash–dotted line). The same applies to the cross-correlation function *r*_{Q20Q30} in the bottom panel. This result recalls Fig. 11 of Part I and confirms that numerical diffusion increases with finer time resolutions, as demonstrated in Part I. The top right panel with *r _{TT}* indicates, however, that the choice

*Dt*= 90 is inappropriate, as the decorrelation time of

*T*′ is much shorter. Overall, the correlation functions support the choice of a grid size

*Dq*= 0.50 and a time step

*Dt*= 30 showing that this EME gives a good description of the dynamics of the studied system.

In Part I, we dealt with a stationary stochastic system; that is, the Lorenz (1963) model with additional white noise, and time series of any length could be easily generated. Figure 4 of Part I reports the value of the ratio *R _{w}* as a function of the time series length; its abscissa spans over four orders of magnitude of length and its right part shows a monotonic decrease of

*R*with growing length for very long time series. This could be interpreted as

_{w}*R*being simply a decreasing function of the number of available observations. In this part of the paper, however, we deal with the real climate, which is a very complex system in continuous evolution and the available data is obtained from limited observational records; as such, the existence of a threshold length beyond which the quality of the EME does not improve cannot be assessed. The ratio

_{w}*R*decreases rapidly with increasing grid size. For example, with

_{w}*Dq*= 1.00 and

*Dt*= 30 we obtain

*R*= 0.14. The impact of time series length on

_{w}*R*is more complicated. By using the first, the middle, and the last 15 years of the available 45-yr-long time series obtained from ERA-40, the ratios

_{w}*R*= 0.35, 0.41, and 0.36, respectively. The last 4.5 yr of the time series lead to a ratio

_{w}*R*= 0.35. For this system, the ratio

_{w}*R*decreases very slowly, yet not monotonically, with increasing time series length. This is somehow similar to the behavior of

_{w}*R*with length for rather short time series observed in the left part of Fig. 4 of Part I and further shows that the ratio

_{w}*R*is a very sensitive quantity.

_{w}Figure 1 gives just a static picture of the distribution of the observed states. To understand more about dynamics in phase space, the EME is integrated from several initial conditions. The initial condition is unit probability in one cell (sharp initial condition). The evolution of the PDF delivered by this EME from an arbitrary initial condition is shown in Fig. 3. The initial positive temperature anomaly abates within the first time step (Figs. 3a and 3b). The high PDF values on the left-hand side of the phase space in Figs. 3b and 3c indicate a QBO East phase between *t* = 120 and *t* = 240. At *t* = 600 the system has developed toward a QBO West phase. Thus, this EME describes the QBO and captures the randomness of transitions between its phases quite well. Some additional mean trajectories are shown in Fig. 4. For each trajectory, a star marks the center of the initial cell and the dots mark the mean position every 30 days. As may be seen on the left panel of Fig. 4, the trajectories show sudden bends after the first time step. Large temperature anomalies abate within the first time step and large positive anomalies reverse at the next time step. In the right panel of Fig. 4, the *T*′ axis has been expanded^{7} to better visualize the behavior of trajectories beyond the time scale of temperature anomalies.^{8} On longer time scales, the circulation in phase space is anticyclonic and contracting toward the origin. The period of the rotation is about 29 months (29 dots in the right panel of Fig. 4). Moreover, the transition from the QBO East phase to the West phase requires approximately 60 days more than the transition from QBO West to QBO East. This difference is consistent with the study of Dunkerton and Delisi (1985, see their Fig. 17). The long-term behavior of a damped oscillator observed in the right panel of Fig. 4 is marked by higher (lower) temperatures during the QBO East (West) phase. This is in qualitative accordance with the results suggested by Holton and Tan (1980).^{9} Furthermore, a slight cooling takes place not only during the transition to QBO West but also during the west phase itself. The inclination of trajectories is approximately −0.03 (nondimensional) from QBO East to QBO West, which corresponds to −0.03 K m^{−1} s; as the trajectories move from QBO West to QBO East, the inclination is approximately −0.05, corresponding to −0.05 K m^{−1} s. This suggests a temperature difference between the two QBO phases of about 2 K. Since the complete deseasonalized data are used, these figures might undergo a seasonal cycle. The standard deviation, *σ*, associated with the PDF is shown in Fig. 5 as a function of the time after the initial condition. After about 500 days, *σ* has grown to 1.3 std dev, which is almost the standard deviation of the data themselves. Most of this study is based on mean trajectories; Fig. 3 and Fig. 5 hint that little insight can be gained by studying higher moments of the probability density distribution.

All in all, the EME provides an interesting visualization of complex dynamics in phase space. It allows an objective, comprehensive, and robust quantification of the relationship between the QBO and the temperature in the Arctic stratosphere. Behind standard approaches like compositing of data there is always the choice of cataloging, for example, winters in QBO East or West (see also Fig. 1 in Thompson et al. 2002); this is not possible when zonal winds in the equator are in between the two phases. The PDF forecasts delivered by the EME are more objective and general. The evolution of the PDF in Fig. 3 and that of the mean trajectories in Fig. 4 describe quite complex behaviors. Such a rich description of dynamics in phase space may not be expected, for instance, from a linear regression model.^{10} The eigenvalues of a linear regression model describe the relations between the variables. With three dimensions, the three eigenvalues are either all real, representing damping rates toward the origin, or there is a real eigenvalue describing a damping rate and two complex conjugate ones describing a damped rotation in phase space. The features described by the EME, including the various trajectory evolutions showing at times sudden bends, cannot be obtained through a linear regression model or by interpreting correlation functions. Moreover, master equations are more general than Fokker–Planck equations (Zwanzig 2001).

Let us turn our attention to a two-dimensional EME for the variable set (*Q*_{30}, *T*′). The phase space grid size is unchanged (*Dq* = 0.50). This EME is derived from the same time series of *Q*_{30} and *T*′ as the three-dimensional one. Fewer cells and consequently fewer transitions need to be estimated in this two-dimensional case and the ratio *R _{w}* = 0.18 (for the three-dimensional case,

*R*was 0.32). The mean trajectory in the left panel of Fig. 6 starts from an initial condition located farther away from equilibrium than in Fig. 3. The PDF forecast after 600 days is shown as well and the square gives the respective mean position. Beyond the decorrelation time of

_{w}*T*′, a monotonic approach to the origin is observed. This two-dimensional EME cannot reproduce the QBO. The right panel of Fig. 6 shows the autocorrelation function

*r*

_{Q30Q30}of the QBO index as estimated from the data (solid line, same as in the top left panel of Fig. 2) and as delivered by the two-dimensional EME (dashed line). The decay of

*r*

_{Q30Q30}as given by the two-dimensional EME confirms that the oscillation between QBO phases is not reproduced and that, given any initial condition, the EME describes a monotonic approach to the origin characterized by strong diffusion. By not reproducing this oscillation, this EME does not capture an important feature of the dynamics of the studied system; caution is needed in conducting a study based on such an EME. Alike the two-dimensional example in section 4d of Part I, a low

*R*value is not a guarantee that the EME gives a good reproduction of the dynamics of the available data. The choice of the initial condition in the left panel of Fig. 6 emphasizes that the mean trajectory approaches the origin along a line with a similar inclination to that observed in Fig. 4; this is a general feature and applies also to trajectories approaching the origin from positive

_{w}*Q*

_{30}values (not shown). This result allows some speculation in section 3b, where two oscillations may not be described because of the limitation to three variables.

### b. The role of the SC

The influence of the QBO and of solar variability on the temperature in the Arctic stratosphere is investigated with an EME for the variable set (*Q*_{30}, *T*′, *S*_{10.7}). It is clear from the discussion above that a satisfactory representation of the QBO and the SC is not possible with this variable set. The inclusion of *Q*_{20} and of a lagged version of *S*_{10.7} as further variables would require a five-dimensional EME. Figure 7a shows a projection on the (*Q*_{30}, *S*_{10.7}) phase plane of this time series distribution of observed states. The appearance of Fig. 7a and the position of the horizontal axis are due to the time series being smooth during solar minimum but very fluctuating elsewhere (see Fig. 4.10 in Dall’Amico 2005). As in section 3a, we partition the phase space with a grid size *Dq* = 0.50, which leads to 410 cells (if *Dq* = 1.00 we obtain only 98 cells). Figure 7b shows isolines of the observed state density (integrated along the *T*′ axis) obtained after a phase space discretization with grid size *Dq* = 0.50. Since all states where *S*_{10.7} < −1 are lumped in the interval −1.5 ≤ *S*_{10.7} < −1.0, the observed state density appears to extend to lower *S*_{10.7} values. The long-term stationary PDF prediction from an EME with the same time step as in section 3a, *Dt* = 30, starting out of an arbitrary initial condition is shown in Fig. 7c. The stationary probability distribution in Fig. 7c is almost identical to the observed state density in Fig. 7b. With *Dq* = 0.50 and *Dt* = 30, the ratio *R _{w}* = 0.55, which is quite high. However, grid size and time step are kept to these values for consistency with the study in section 3a. The two maxima in Figs. 7b and 7c are a consequence of the distribution of the values of

*Q*

_{30}. The slight local maximum centered at (−1.25, 0.25) and the minimum on its lower right seem to hint at possibly different dynamics for low and high solar activity. The mean trajectory from the chosen initial condition is shown as well in Fig. 7c. Beyond an initial increase in the value of

*S*

_{10.7}, this trajectory approaches the origin without any oscillation.

The solid lines in Fig. 8 give some sample correlation functions for this variable set. The top panels report the autocorrelation functions. New with regards to Fig. 2 is *r*_{S10.7S10.7} (thin solid line), whose minimum occurs at a lag of 5 yr and 2 months. (Its next maximum occurs at a lag of 10 yr and 5 months, not shown.) The bold solid line in the bottom panel describes *r*_{S10.7Q30}; its negative value for lag zero and its minimum for positive lags of about 5 months suggest that during solar maximum easterly anomalies of *Q*_{30} are favored (in agreement with e.g., Labitzke 2001). The thin solid line in the same panel is *r*_{S10.7T′}; at lag zero, *r*_{S10.7T′} is positive and grows beyond 0.1 when *S*_{10.7} leads *T*′ by a few months. A QBO-like frequency is not immediately visible in *r*_{S10.7T′}. However, the minima and maxima of *r*_{S10.7Q30} tend to coincide with (or precede by up to a couple of months) local changes of the opposite sign of *r*_{S10.7T′}. Though quite small in its extent, this feature suggests that local increases (decreases) in *T*′ can be expected when the SC and the QBO have the opposite (same) sign and is in agreement with other studies (e.g., Labitzke 1987). The dashed lines in Fig. 8 show the corresponding correlation functions as delivered by the EME. In the top left panel are reported *r*_{Q30Q30} (bold dashed line) and *r*_{S10.7S10.7} (thin dashed line); both indicate that this EME describes a monotonic approach to the origin. The inability to follow the oscillations is evident also in the evolution of the dashed lines in the bottom panel. Caution is needed in evaluating results from this EME since it cannot reproduce the two observed oscillations. The decay of *r*_{Q30Q30} as delivered by this EME (bold dashed line, top left panel) recalls the decay of *r*_{Q30Q30} delivered by the two-dimensional EME for (*Q*_{30}, *T*′) in Fig. 6b. To reproduce both the QBO and the SC, we should derive a five-dimensional EME for *Q*_{30}, *T*′, *S*_{10.7}, *Q*_{20}, and a last variable describing, for instance, the value of *S*_{10.7}, say, 4 yr before. If adequate computer resources and observational records were available, we could expect such an EME to deliver better adherent correlation functions and to allow a much deeper insight in the underlying dynamics. We extend here the considerations made at the end of section 3a while comparing the two-dimensional with the three-dimensional EME in order to speculate on what the PDF evolution would look like in the full five-dimensional case.^{11}

Regarding the length of the time series, its first, middle, and last 15 yr lead with *Dq* = 0.50 and *Dt* = 30 to ratios *R _{w}* = 0.61, 0.56, and 0.61, respectively. The last 4.5 yr of the time series lead to

*R*= 0.63. This is a situation in which a longer time series would be desirable.

_{w}Several EME runs are started from cells where the state vector (*Q*_{30}, *T*′, *S*_{10.7}) resides 30 or more times within the time series, so the estimates of the transition coefficients for the first time steps have narrower confidence intervals. The symbols in Figs. 9, 10, 11 and 12 give the mean position every 30 days and can be used to identify a trajectory while comparing a left panel with the corresponding right panel. The left panels show projections on the (*Q*_{30}, *S*_{10.7}) phase plane. The right panels show projections on the (*Q*_{30}, *T*′) phase plane. The *T*′ axis is expanded for the same reasons and to allow direct comparisons with the right panel in Fig. 4. The inclination of trajectories in the right panels of Figs. 9, 10, 11 and 12 during their approach to the origin can be compared with the inclinations in the right panel of Fig. 4. Figure 9 shows selected mean trajectories that begin during a QBO East phase and solar maximum. Figure 10 shows selected mean trajectories that begin during a QBO West phase and solar maximum. With some abstraction and recalling the quantitative discussion made on Fig. 4 and Fig. 6, we may interpret the dynamics observed during high solar activity in Fig. 9 and Fig. 10. The relationship seen in the right panel of Fig. 4 is shifted here to higher temperature anomalies of +0.1 std dev (corresponding to +1 K) and more. Without a five-dimensional EME, it is hard to interpret the surprisingly high steepness in the (*Q*_{30}, *T*′) phase plane of the trajectory marked by the diamonds in Fig. 9. Trajectories beginning with either *Q*_{30} = 0.25 or *S*_{10.7} = −0.25 at times display behavior exhibited for lower or higher values of the respective variables. Let us consider trajectories beginning with low *S*_{10.7} values. Figure 11 shows selected mean trajectories that begin during a QBO East phase and solar minimum. Figure 12 shows selected mean trajectories that begin during a QBO West phase and solar minimum. The relationship between *Q*_{30} and *T*′ discussed in section 3a is shifted in Figs. 11 and 12 toward lower temperature anomalies of −0.1 std dev (corresponding to −1 K) and less. Further, a comparison of the left with the right panels of Figs. 9, 10, 11 and 12 reveals that the inclination of a trajectory as projected onto the left panel, relative to other trajectories, determines the relative inclination of its approach to the origin on the right panel. All in all, this three-dimensional EME shows that *T*′ is increased by high solar activity and that this, though to a different extent, occurs during both phases of the QBO. These results agree with the study of Crooks and Gray (2005, see their Fig. 2) and with the description of Lean (2005), but are somewhat different from the ones proposed by some other studies (e.g., Labitzke 2001, 2004; Salby and Callaghan 2002). A further study of these differences goes beyond the scope of this paper. The amplitudes of the response of *T*′ to the QBO and the SC are similar (∼1 K), yet because of the mismatch between the correlation functions delivered by the EME from those obtained from the data, the interpretation of Figs. 9, 10, 11 and 12 is to some extent speculative. However, there is an indication of an influence of the QBO and of the SC on *T*′, which is well established in the whole phase space.

### c. The NAM in the stratosphere and in the troposphere

The dynamical coupling of stratosphere and troposphere in NH midlatitudes is investigated with an EME for the variable set (*A*_{10}, *A*_{100}, *A*_{850}). Figures 13a and 13d show projections of this time series distribution of observed states on the (*A*_{10}, *A*_{100}) and on the (*A*_{10}, *A*_{850}) phase planes, respectively. The correlation between *A*_{10} and *A*_{100} is strong and positive. The connection between *A*_{10} and *A*_{850} is also positive but much weaker. The variability increases at levels closer to the ground. This is also suggested by comparing the time series of these variables (not shown, see Fig. 4.17, 4.18, and 4.19 in Dall’Amico 2005), where in several cases persistent NAM anomalies are observed at the different pressure levels, but it is hard to identify whether these anomalies propagate preferentially upward or downward. If the grid size is set at *Dq* = 0.50, the data cloud is partitioned into 456 cells. With *Dq* = 0.50 and with the available time series, the ratio *R _{w}* would be quite high; that is, 0.4 <

*R*< 0.9 depending on the time step

_{w}*Dt*, with

*R*= 0.74 for

_{w}*Dt*= 7 days. Therefore, a grid size

*Dq*= 1.00 is chosen in order to increase the statistical significance of the transition coefficient estimates. With

*Dq*= 1.00, the data cloud is discretized into 105 cells. Further, the time series and the correlation functions (see Fig. 14) suggest that there is little memory beyond a few weeks time, which would suggest a time step of

*Dt*= 7 days. However, in order to reduce the more rapid smearing of the PDF connected with the coarser grid size, we discuss here results obtained with a time step

*Dt*= 14. The ratio

*R*obtained with

_{w}*Dq*= 1.00 and

*Dt*= 14 is an acceptable

*R*= 0.32. Figures 13b and 13e show isolines of this time series observed state density (integrated along the respective perpendicular axis) for the chosen grid size

_{w}*Dq*= 1.00 and time step

*Dt*= 14. The long-term, stationary PDF predicted by an EME with the chosen grid size and time step for an arbitrary initial condition is shown in Figs. 13c and 13f, and is almost identical to Figs. 13b and 13e. The trajectory described by the mean position is shown as well. Moreover, Fig. 13 indicates that

*A*

_{10}is more closely linked to

*A*

_{100}than to

*A*

_{850}. Analogous projections on the (

*A*

_{100},

*A*

_{850}) phase plane indicate that

*A*

_{100}is better linked to

*A*

_{10}than to

*A*

_{850}(not shown).

The solid lines in the panels of Fig. 14 report the correlation functions as estimated from the data as a function of time lag. The values of the cross-correlation functions for negative (positive) lags are related to a downward (upward) propagation of NAM anomalies. For *r*_{A100,A10} and *r*_{A850,A10}, the correlations are much higher for negative lags where their decrease begins with a shoulder at a negative lag of about 20 days, suggesting downward propagation. A more symmetric picture is given by *r*_{A850,A100}, which also shows slightly higher values for positive lags of a few days. Ambaum and Hoskins (2002) discussed correlation functions for a NAO index and an index measuring the strength of the stratospheric polar vortex estimated from data from the winter months. They found a small peak in the cross-correlations for the NAO index leading the stratospheric one by 4 days. It is, however, difficult to identify clear time scales on the basis of the results in Fig. 14 as the correlations do not show other peaks besides the one next to lag zero. Only *r*_{A850,A10} shows two flattened maxima for negative lags of 5 to 20 days. The dashed lines in Fig. 14 report the correlation functions as delivered by the EME. The EME captures the decay of the autocorrelation functions (top panels) and the fact that this decay is more rapid at 100 and 850 hPa. The cross-correlation functions are also well reproduced by the EME. All in all, this EME well reproduces the dynamics of the studied system.

For the chosen grid size, *Dq* = 1.00, and time step, *Dt* = 14, the ratios *R _{w}* for the first, the middle, and the last 15 yr of the time series are 0.42, 0.43, and 0.44, respectively. The last 4.5 yr of the time series lead to a ratio

*R*= 0.58. The underlying process is thus quite stationary and the available time series is sufficiently long. The latter is no surprise as the decorrelation times observed in Fig. 14 are much shorter than the ones observed in Fig. 2 and Fig. 8.

_{w}EME runs are started from all cells with 30 or more observations. Figure 15 shows projections of the mean trajectories delivered by these runs on the (*A*_{10}, *A*_{100}) phase plane. The same trajectories are projected in Fig. 16 on the (*A*_{10}, *A*_{850}) phase plane and in Fig. 17 on the (*A*_{100}, *A*_{850}) phase plane. Use is made of different symbols which give the mean position every *Dt* = 14 days in order to recognize any trajectory in the projections. Since in the figures the three-dimensional phase space is projected onto a coordinate plane, in some cases more than one trajectory appears to begin from the same initial condition. Moreover, since the phase space has been discretized, only cell centers can be used as initial conditions. The right panels in Figs. 15 –17 zoom in by a factor of 10 toward the origin. If *A*_{10} and *A*_{100} were just slightly related to another and showed approximately the same time scales, one would expect the trajectories in Fig. 15 to evolve along a straight line toward the origin. As *A*_{10} has a slightly longer memory than *A*_{100}, one would expect two curves recalling, say, parabolas with vertical axes and vertexes at the origin. What one observes, however, is a clear cyclonic rounding of the contracting motion. Moreover, in both panels in Fig. 15 and even more clearly in the right one, one notes that trajectories starting close to the *A*_{10} axis exhibit a stronger cyclonic rounding than those starting close to the *A*_{100} axis. This suggests that anomalies of *A*_{10} propagate downward to 100 hPa with a time scale of approximately 14 (or more) days, that is one (at times two) point(s) on the trajectories. This time scale is suggested also by runs with an EME with *Dq* = 0.50 and *Dt* = 7 (not shown).

The same trajectories discussed in the preceding paragraph are projected onto the (*A*_{10}, *A*_{850}) phase plane in Fig. 16. Here the cyclonic rounding of trajectories is less pronounced and can be better appreciated in the right panel. This is probably an effect of the increasing distance between these layers, as 10 hPa correspond to a height above sea level of approximately 30 km and 850 hPa to about 1.5 km. The typical time scale for the propagation of a NAM anomaly from 10 hPa to 850 hPa is approximately 4 weeks (two points in Fig. 16). This time scale is confirmed by runs with *Dq* = 0.50 and *Dt* = 7 (not shown). Since 850 hPa correspond to the top of the planetary boundary layer over land in midlatitudes, the rounding of the trajectories in Fig. 16 implies that anomalies of *A*_{10} propagate deep into the troposphere. These results are consistent with other studies (e.g., Baldwin and Dunkerton 1999; Black 2002; Baldwin et al. 2003). The time scales suggested by the EME are consistent with those suggested by other studies for the winter season (the trajectories represent the evolution of a statistical mean); for example, Baldwin and Dunkerton (1999) suggested a time scale of about three weeks for the propagation of such anomalies from 10 hPa to the surface. Figure 2 of Baldwin and Dunkerton (2001) hints at a time scale of two weeks and shows a long persistence of the anomalies up to the tropopause.

These first results somehow suggest that tropospheric NAM anomalies hardly influence the stratosphere. In Fig. 17, the trajectories are projected on the (*A*_{100}, *A*_{850}) phase plane. Dynamics appear more vivid when projected on this phase plane, suggesting a strong interaction. Moreover, a clear cyclonic behavior does not appear. The impact of large tropospheric NAM anomalies to the lower stratosphere (slight anticyclonic rounding) is generally confined to the first two segments of a trajectory. However, the trajectories have no anticyclonic rounding in their approach to the origin. These results seem to support speculation that the troposphere affects the stratosphere on shorter time scales (e.g., Ambaum and Hoskins 2002).

## 4. Conclusions

EMEs are constructed directly from data and provide a statistical model of a system in terms of the PDF. They can capture nonlinear behavior. The significance of the transition coefficient estimates has been assessed in terms of a weighted-average ratio, *R _{w}*, between the half-width of the confidence intervals of the estimates and the estimates themselves. The adherence of correlation functions as delivered by the EME to those obtained from the data has been used as a test of how well the EME reproduces the dynamics of the system. EMEs have been derived from time series of climate variables representing various modes of stratospheric climate variability. The time series have been obtained from ERA-40 daily means and from observations and have been normalized and deseasonalized. The transition coefficients have been estimated from the daily time series on the basis of time steps of up to 90 days. Following the guidelines developed in Part I, any phase space has been partitioned into a few hundred cells and in the case of the NAM the grid size was subsequently increased to allow for an acceptable ratio

*R*. Since the time series used in this part of the paper come from reanalysis data and observations that are limited in time, the existence of a threshold length beyond which the quality of the PDF forecasts hardly improves, as was found in Part I, cannot be assessed. The EMEs derived in this part of the paper confirm the remaining results of Part I. EMEs with coarser time steps show less numerical diffusion and deliver improved correlation functions. The dimensionality has proven to be an important constraint.

_{w}A three-dimensional EME has first been derived from the time series of variables for the QBO at 30 hPa, *Q*_{30}, and 20 hPa, and of the temperature anomaly in the Arctic middle stratosphere, *T*′. The evolution of the PDF in the phase space of these variables as predicted by the EME reproduces the probabilistic character of the transition from one QBO phase to the other. The transition from a QBO West to a QBO East phase takes about 2 months more than the transition from East to West. A straightforward plot of the data, which cover the whole period between September 1957 and August 2002, does not uniformly show the relationship suggested by Holton and Tan (1980) on the basis of monthly means for NH winter months. However, the evolution of the PDF shows that during the QBO East phase the Arctic middle stratosphere is approximately 2 K warmer than during the QBO West phase. This does confirm and quantify the Holton and Tan relationship. A two-dimensional EME derived from the time series of *Q*_{30} and *T*′ cannot describe the oscillation associated with the QBO, but its comparison with the previous one has helped to interpret the results of another EME derived from the time series of *Q*_{30}, *T*′, and of the variable for the solar flux at a wavelength of 10.7 cm. This EME hints that the relationship between the QBO and the temperature in the Arctic stratosphere discussed in the previous paragraph is shifted toward warmer (colder) temperatures of about +1 (−1) K during phases of high (low) solar activity. However, the correlation functions delivered by this EME suggest that a five-dimensional EME is needed to better capture the underlying dynamics.

Finally, an EME has been derived from the time series of variables representative of the NAM in the middle and lower stratosphere and in the lower troposphere. This EME shows that NAM anomalies tend to propagate downward from the middle stratosphere into the lower stratosphere in approximately two weeks and then deeply into the troposphere with a time scale of about four weeks. The upward impact of large tropospheric NAM anomalies on the stratosphere is confined to the lower stratosphere.

## Acknowledgments

The results reported in these papers derive from a doctoral thesis (Dall’Amico 2005). We wish to thank Prof. Klaus X. Elsässer, Prof. M. Dameris, and many colleagues at the Meteorological Institute of the University of Munich who spent time in fruitful discussions with us. In particular, discussions with the late Dr. Wolfgang Ulrich were helpful. We thank Dr. Chris Ferro for his statistical advice. Several suggestions by Dr. Katie Coughlin, Prof. Lesley J. Gray, and anonymous reviewers helped to improve this paper. We thank the ECMWF for the development of ERA-40 and the Herberg Institute of Astrophysics, Canada, for making the time series of the 10.7-cm solar flux available on the internet. Financial support was provided by the German Ministry of Education and Research and the German Aerospace Center within KLIMESTO, a project of the German Climate Research Program, Contract 01LD0033.

## REFERENCES

Ambaum, M. H. P., , and B. J. Hoskins, 2002: The NAO troposphere–stratosphere connection.

,*J. Climate***15****,**1969–1978.Baldwin, M. P., , and T. J. Dunkerton, 1999: Propagation of the Arctic Oscillation from the stratosphere to the troposphere.

,*J. Geophys. Res.***104****,**30937–30946.Baldwin, M. P., , and T. J. Dunkerton, 2001: Stratospheric harbingers of anomalous weather regimes.

,*Science***294****,**581–584.Baldwin, M. P., and Coauthors, 2001: The quasi-biennial oscillation.

,*Rev. Geophys.***39****,**179–230.Baldwin, M. P., , D. B. Stephenson, , D. W. J. Thompson, , T. J. Dunkerton, , A. J. Charlton, , and A. O’Neill, 2003: Stratospheric memory and skill of extended-range weather forecasts.

,*Science***301****,**636–640.Black, R. X., 2002: Stratospheric forcing of surface climate in the Arctic Oscillation.

,*J. Climate***15****,**268–277.Cencini, M., , G. Lacorata, , A. Vulpiani, , and E. Zambianchi, 1999: Mixing in a meandering jet: A Markovian approximation.

,*J. Phys. Oceanogr.***29****,**2578–2594.Coughlin, K., , and K-K. Tung, 2001: QBO Signal found at the extratropical surface through northern annular modes.

,*Geophys. Res. Lett.***28****,**4563–4566.Crommelin, D. T., 2004: Observed nondiffusive dynamics in large-scale atmospheric flow.

,*J. Atmos. Sci.***61****,**2384–2396.Crooks, S. A., , and L. J. Gray, 2005: Characterization of the 11-year solar signal using a multiple regression analysis of the ERA-40 dataset.

,*J. Climate***18****,**996–1015.Dall’Amico, M., 2005: Data-based master equations for the stratosphere. Ph.D. thesis, Ludwig-Maximilians-Universität of Munich, 71 pp. [Available online at http://edoc.ub.uni-muenchen.de/archive/00003890/.].

Dall’Amico, M., , and J. Egger, 2007: Empirical master equations. Part I: Numerical properties.

,*J. Atmos. Sci.***64****,**2981–2995.Dunkerton, T. J., , and D. P. Delisi, 1985: Climatology of the equatorial lower stratosphere.

,*J. Atmos. Sci.***42****,**376–396.Egger, J., 1996: Comments on “On the ‘downward control’ of extratropical diabatic circulations by eddy-induced mean zonal forces.”.

,*J. Atmos. Sci.***53****,**2103–2104.Egger, J., 2001: Master equations for climatic parameter sets.

,*Climate Dyn.***18****,**169–177.Fraedrich, K., 1988: El Niño–Southern Oscillation predictability.

,*Mon. Wea. Rev.***116****,**1001–1012.Gray, L. J., , S. Crooks, , C. Pascoe, , S. Sparrow, , and M. Palmer, 2004: Solar and QBO influences on the timing of stratospheric sudden warmings.

,*J. Atmos. Sci.***61****,**2777–2796.Haynes, P. H., 2005: Stratospheric dynamics.

,*Annu. Rev. Fluid Mech.***37****,**263–293.Haynes, P. H., , C. J. Marks, , M. E. McIntyre, , T. G. Shepherd, , and K. P. Shine, 1991: On the “downward control” of extratropical diabatic circulations by eddy-induced mean zonal forces.

,*J. Atmos. Sci.***48****,**651–678.Haynes, P. H., , M. E. McIntyre, , and T. G. Shepherd, 1996: Reply.

,*J. Atmos. Sci.***53****,**2105–2107.Holton, J. R., , and H-C. Tan, 1980: The influence of the equatorial quasi-biennial oscillation on the global circulation at 50 mb.

,*J. Atmos. Sci.***37****,**2200–2208.Houghton, J. T., , Y. Ding, , D. J. Griggs, , M. Noguer, , P. J. van der Linden, , X. Dai, , K. Maskell, , and C. A. Johnson, 2001:

*Climate Change 2001: The Scientific Basis*. Cambridge University Press, 881 pp.Labitzke, K., 1987: Sunspots, the QBO, and the stratospheric temperature in the north polar region.

,*Geophys. Res. Lett.***14****,**535–537.Labitzke, K., 2001: The global signal of the 11-year sunspot cycle in the stratosphere: Differences between solar maxima and minima.

,*Meteor. Z.***10****,**83–90.Labitzke, K., 2004: On the signal of the 11-year sunspot cycle in the stratosphere and its modulation by the quasi-biennial oscillation.

,*J. Atmos. Sol. Terr. Phys.***66****,**1151–1157.Labitzke, K., , and H. van Loon, 1999:

*The Stratosphere: Phenomena, History, and Relevance*. Springer, 179 pp.Lean, J., 2005: Living with a variable sun.

,*Phys. Today***58****,**32–38.Lorenz, E. N., 1951: Seasonal and irregular variations of the Northern Hemisphere sea-level pressure profile.

,*J. Meteor.***8****,**52–59.Lorenz, E. N., 1963: Deterministic nonperiodic flow.

,*J. Atmos. Sci.***20****,**130–141.Pasmanter, R. A., , and A. Timmermann, 2002: Cyclic Markov chains with an application to an intermediate ENSO model.

,*Nonlinear Proc. Geophys.***10****,**197–210.Qian, B., , J. Corte-Real, , and H. Xu, 2000: Is the North Atlantic Oscillation the most important atmospheric pattern for precipitation in Europe?

,*J. Geophys. Res.***105****,**11901–11910.Reed, R. J., , W. J. Campbell, , L. A. Rasmussen, , and D. G. Rogers, 1961: Evidence of the downward-propagating annual wind reversal in the equatorial stratosphere.

,*J. Geophys. Res.***66****,**813–818.Rind, D., 2002: The sun’s role in climate variations.

,*Science***296****,**673–677.Salby, M., , and P. Callaghan, 2000: Connection between the solar cycle and the QBO: The missing link.

,*J. Climate***13****,**328–338.Salby, M., , and P. Callaghan, 2002: Evidence of the solar cycle in the general circulation of the stratosphere.

,*J. Climate***17****,**34–46.Schönwiese, C. D., 1985:

*Praktische Statistik für Meteorologen und Geowissenschaftler*. 2d ed. Gebrüder Borntraeger, 231 pp.Spekat, A., , B. Heller-Schulze, , and M. Lutz, 1983: Über Großwetter und Markov-Ketten (“Großwetter” circulation analysed by means of Markov chains).

,*Meteor. Rundsch.***36****,**243–248.Stolarski, R. S., , P. Bloomfield, , R. D. McPeters, , and J. R. Herman, 1991: Total ozone trends deduced from Nimbus 7 TOMS data.

,*Geophys. Res. Lett.***18****,**1015–1018.Thompson, D. W. J., , and J. M. Wallace, 1998: The Arctic Oscillation signature in wintertime geopotential height and temperature fields.

,*Geophys. Res. Lett.***25****,**1297–1300.Thompson, D. W. J., , and J. M. Wallace, 2000: Annular modes in the extratropical circulation. Part I: Month-to-month variability.

,*J. Climate***13****,**1000–1016.Thompson, D. W. J., , and J. M. Wallace, 2001: Regional climate impacts of the Northern Hemisphere annular mode.

,*Science***293****,**85–89.Thompson, D. W. J., , M. P. Baldwin, , and J. M. Wallace, 2002: Stratospheric connection to Northern Hemisphere wintertime weather: Implications for predictions.

,*J. Climate***15****,**1421–1428.Uppala, S. M., and Coauthors, 2005: The ERA-40 re-analysis.

,*Quart. J. Roy. Meteor. Soc.***131****,**2961–3012.Veryard, R. G., , and R. A. Ebdon, 1961: Fluctuations in tropical stratospheric winds.

,*Meteor. Mag.***90****,**125–143.Wallace, J. M., 2000: North Atlantic Oscillation/annular mode: Two paradigms—One phenomenon.

,*Quart. J. Roy. Meteor. Soc.***126A****,**791–805.Zwanzig, R., 2001:

*Nonequilibrium Statistical Mechanics*. Oxford University Press, 222 pp.

^{1}

The issue of stratosphere–troposphere exchange, highly relevant in topics concerned with atmospheric chemistry, is not addressed here.

^{2}

Effort has been made in making this part of the paper accessible to the reader who is unaware of this technique and has not read Part I. However, it is recommended to read section 2 of Part I first.

^{3}

Depending on the user’s aims, various quantities may be derived from the EME, like velocity and diffusion fields in phase space (Egger 2001), eigenvalues and eigenvectors (Egger 2001; Pasmanter and Timmermann 2002), and entropy production (Pasmanter and Timmermann 2002).

^{4}

The derivation of EMEs for a single season is quite difficult, as the transition coefficients have to be estimated on the basis of “chips” of a long time series. The data have to be manually processed in order to obtain a stationary solution for the EME and this solution may differ substantially from the observed state density (defined in Part I and recalled in the next section). In the stationary solution of the EME derived from ERA-40 monthly daily means for the winter months only, the QBO west phase is highly overemphasized.

^{5}

For an EME derived from monthly daily means of the variables, the ratio *R _{w}* = 0.85. This supports the choice of working with daily means and large ϒ. The correlation functions delivered by the EME for ERA-40 monthly daily means have, however, a similar progression to those discussed below for ERA-40 daily means with

*Dt*= 30 (not shown).

^{6}

Owing to the volume of the data (16 436 data points), even very low correlations are significant. A *t* test (see Schönwiese 1985, 138–143) suggests that a value of *r* = 0.02 is significant at the 99% confidence level for lags up to 1500 days. It could be argued that the data are not completely independent. The number of data could be divided by the number of days needed for *r*_{Q30Q30} to decay to *e*^{−1} to obtain approximately 100 independent data. In this case a value *r* = 0.15 would still be significant at the 90% confidence level.

^{7}

The variance of a mean coordinate is inversely proportional to the number of independent observations. For a description like Fig. 1b there is the complicating fact that observations are lumped into (the centers of) cells. Once the transition matrix is estimated from the available time series, there is no further sampling variability associated with the mean coordinates at each time step. If time series of the variables were accessible for longer periods than that of ERA-40, a different behavior may be revealed, but the dynamics illustrated in the right panel of Fig. 4 are compatible with the equilibrium state. The trajectories arise from deterministic quantities given by the mean coordinates at each time step. Therefore, overall statements can be made based on the details revealed by the right panel of Fig. 4.

^{8}

In Fig. 4, points are plotted and connected with lines. Parts of these lines do not show if the points at their extrema are both outside the range of the plot, so that parts of the trajectories in the left panel of Fig. 4 do not show in the right panel.

^{9}

Holton and Tan (1980) found in the zonal mean at 50 hPa for November–December a geopotential height difference of 100 m north of 75°N, and a geostrophic wind difference of 5 m s^{−1} at 65°N.

^{10}

A linear regression model was compared by Egger (2001) to a two-dimensional EME.

^{11}

In the case of a large number of similar variables, it is natural to derive an EME from the principal components of the main EOFs. For the QBO and SC, however, the interpretation of an EME for, say, the principal components of the first three EOFs of the five-dimensional variable set would be very difficult.