## 1. Introduction

In this paper, the influence of stationary waves on the maintenance of the leading variability patterns of the extratropical troposphere is investigated. The low-frequency variability of the tropospheric circulation includes a characteristic zonal-mean component, which consists of a north–south oscillation of the zonal-mean zonal wind. This behavior is described either by the zonal index (Rossby et al. 1939) via the variability pattern of the zonal-mean zonal wind or, including the longitudinal dependence, by the annular modes (AMs) of the geopotential at different pressure levels (Thompson and Wallace 1998, 2000; Gong and Wang 1999). While the southern AM is approximately zonally symmetric, the northern AM includes a planetary wave structure that reflects the regional variability pattern of the North Atlantic Oscillation (NAO; Walker and Bliss 1932; van Loon and Rogers 1978). The high phase of the AM variability patterns with a positive value in its associated time series means a poleward shift of the tropospheric westerly jet from its climatological position, while the low phase with a negative value means an equatorward shift. The physical relevance of the zonal index and of the AMs is disputable (Wallace and Hsu 1985; Ambaum et al. 2001; Ting et al. 2000; Christiansen 2002; Wallace and Thompson 2002). Nevertheless, owing to the high correlation between the time series of the zonal index, the northern AM and the NAO, Wallace (2000) argues that the patterns are different statistical interpretations of the same entity.

The variability patterns of the zonal-wind oscillation result from wave–mean flow interaction, as several simulations with atmospheric general circulation models have shown (Glowienka-Hense 1990; Robinson 1991; Limpasuvan and Hartmann 2000). From observational data, Karoly (1990) demonstrates for the zonal index of the Southern Hemisphere that the zonal wind is associated with transient eddy fluxes that maintain the wind anomaly. Similarly, Hurrell (1995) finds that the transient vorticity fluxes due to high-frequency waves with periods of 2 to 8 days reinforce the upper-tropospheric streamfunction anomalies connected with the NAO. Recently, Lorenz and Hartmann (2001, 2003) have performed a time series analysis of the vertically integrated angular momentum budget finding that the observed AMs, both of the northern and the southern winter hemisphere, are mainly determined by the feedback between the zonal-mean zonal wind and high-frequency eddies. Robinson (2000) has suggested a corresponding feedback mechanism, where strong surface friction acting on the anomalous barotropic westerlies creates a zone of enhanced baroclinicity, and where the meridional propagation of the resulting baroclinic eddies reinforces the westerlies through their Eliassen–Palm flux convergence.

Besides the meridional displacement of the westerlies, the northern AM is characterized by strong anomalous^{1} stationary waves. These waves are related to the zonal-mean zonal-wind fluctuations of the zonal index (Ting et al. 1996; DeWeaver and Nigam 2000a). The basic mechanism is linear zonal–eddy coupling; that is, the anomalous zonal-mean westerlies modulate the linear propagation of planetary waves and thereby induce anomalous stationary waves. In the vertically integrated framework, the coupling of the anomalous and climatological stationary waves yields an anomalous meridional eddy momentum flux convergence that feeds back on the anomalous zonal-mean westerlies associated with the NAO (DeWeaver and Nigam 2000b). Using National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data, Limpasuvan and Hartmann (2000) have shown that, during the phases of the northern AM, the anomalous stationary waves indeed provide the largest contribution to the eddy momentum flux. In addition, Lorenz and Hartmann (2003) point out that the eddy fluxes due to the anomalous stationary waves produce a northward shift of the AM pattern. On the other hand, Thompson et al. (2003) argue that transient eddies alone maintain the northern AM, while the apparent influence of the anomalous stationary waves on the northern AM possibly reflects the effect of the transient eddies on the stationary waves. These different conclusions indicate that the importance of the anomalous stationary waves for the northern AM has yet to be clarified.

The focus of the present work is to investigate the role of baroclinic waves and anomalous stationary waves in the maintenance of the AM. Since both time scales are coupled in the observational data of the Northern Hemisphere, we approach this question by using a simple general circulation model (GCM) with different stationary wave forcings. Similar models have been used before in order to study the influence of idealized stationary wave forcing on the NAO and AM. So, Franzke et al. (2001) have demonstrated by employing two heating dipoles in a simple GCM that both the interaction between stationary waves and the low-frequency flow and the self-interaction among the high-frequency eddies can amplify an NAO-like variability pattern. Taguchi and Yoden (2002) have found a strong structural dependence of the AM on the height of an idealized orography. We apply now more realistic stationary wave forcing by orography and land–sea heating contrasts. In the different model experiments, the stationary wave forcing ranges from the Northern Hemispheric realistic case, over two intermediate cases with either orography or land–sea heating contrasts, to an aquaplanet simulation with axisymmetric boundary conditions. Previously, Körnich et al. (2003) have shown that these experiments exhibit AM-like variability patterns in the upper troposphere, but with different strengths and longitudinal structures. Based on the same experiments, we shall investigate here the feedback processes of the AM that result from the coupling, on the one hand, with the baroclinic eddies, and on the other, with anomalous stationary waves. The time scales of the baroclinic and stationary waves are separated with appropriate time filtering. To estimate the contribution of the different waves to the maintenance of the AM, we use the time series analysis of the vertically integrated zonal momentum budget after Lorenz and Hartmann (2001, 2003). The feedback process between the anomalous stationary waves and the AM (DeWeaver and Nigam 2000b) is further examined with a linear wave model, where the zonal-mean flow during the AM phases is taken as the basic state. From the linearly calculated stationary waves we determine the respective eddy momentum flux convergence, and hence the feedback on the zonal-mean zonal wind. Our main result is that only the combination of the orography and the land–sea heating contrasts leads to the feedback between the anomalous stationary waves and the zonal-mean zonal wind, and thus to a realistic simulation of the northern AM. This result contributes to a better understanding of the hemispherical differences of the AM, and it has implications for the dependence of the northern AM on climate changes of the stationary waves.

The model data with its respective climatologies are presented in the sections 2a and 2b. Section 2c gives a description of the linear model version and the analysis methods. The leading variability patterns of the lower troposphere are shown in section 3a. Regarding the zonal momentum budget, the temporal behavior of the stationary and baroclinic waves is examined in section 3b. The feedback between the anomalous stationary waves and the AM is further studied on the basis of a composite analysis of the AM and with a linear model of the stationary waves (section 3c). Finally, the results are discussed in section 4 and summarized in section 5.

## 2. Model and methods

### a. Circulation model

*T*. The temperature profile

_{E}*T*is adapted to Northern Hemispheric January conditions, and all simulations are equivalent “perpetual January” runs. The relaxation time

_{E}*τ*is 16 days in the troposphere and drops to 4 days in the uppermost model layers. Longitudinally dependent conditions due to land–sea heating contrasts are introduced into the model with a prescribed cumulus heating in the deep Tropics (

*Q*), with the dynamically induced condensational heating in middle latitudes using a heating function

_{c}*Q*, and with the associated convergence

_{m}*q*of the vertical sensible heat flux. The latter is defined via a standard boundary layer scheme using the surface temperature

*T*= [

_{s}*T*+ 0.4

_{E}*τ*(

*Q*+

_{c}*Q*)]

_{m}_{surface}. The heating rate of the midlatitudinal condensational heating scales with the pressure velocity

*ω*, but for upward motion only due to the Heaviside function

*ω*). The remaining terms on the rhs of Eq. (1) are horizontal diffusion of temperature

*H*and frictional heating

_{T}*ϵ*. Horizontal diffusion is applied in the form of a ∇

^{2}scheme in order to ensure the frictional heating to be positive definite (Becker 2001). The kinematic viscosity amounts 5.1 × 10

^{4}m

^{2}s

^{−1}in the troposphere and increases to 4.0 × 10

^{6}m

^{2}s

^{−1}in the lower mesosphere. Additional information on the model is documented by Becker and Schmitz (2001) and Becker (2003).

Stationary waves are forced either orographically or thermally. Orographic forcing consists of an envelope world orography Φ* _{s}* with a maximum elevation of 4.4 km in the vicinity of the Himalayas (Fig. 1a). The global mean of the orography is 236.7 m, which agrees well with the observational value of 237.3 m (Joseph 1980). The heating functions

*Q*and

_{c}*Q*are prescribed functions of longitude, latitude, and pressure (Figs. 1b–d). The functions are tuned such that the resulting diabatic heating [Eq. (1)] matches with observational analysis for Northern Hemispheric winter conditions (Wang and Ting 1999).

_{m}In the different model experiments, the stationary wave forcings Φ* _{s}*,

*Q*, or

_{m}*Q*are taken into account in different combinations, while the zonal-mean components of the forcing fields are the same in all experiments. Table 1 shows the model experiments with the respective combinations of stationary wave forcing. Time integration is performed with Δ

_{c}*t*= 12 min for 18 001 days in each experiment. Experiment AQUA is our control experiment without stationary wave forcing. Two simulations are considered with either orographic (experiment ORO) or thermal forcing of stationary waves (experiment TH).

^{2}Finally, experiment FULL includes the full stationary wave forcing representing the wintertime Northern Hemisphere. Note that our separation of orographic and thermal stationary wave forcing is artificial, but it allows us to assess the influence of individual forcings on the annular modes, for example, land–sea heating contrasts in the absence of a mountain torque, because the interaction between the different forcings is less important than the interaction between the forced wave fields (Held 1983).

### b. Model climate

Owing to the different stationary wave forcing, the model experiments exhibit different climatological states that yield the background for the variability patterns. Despite the model’s extension into the stratosphere we focus here on the tropospheric state. Figure 2 shows the time-averaged zonal-mean zonal wind. In experiment AQUA the wind reaches values up to 55 m s^{−1} in winter. The introduction of stationary wave forcing in the other experiments induces a weakening of the mean zonal wind around 40°N (Figs. 2b–d). This happens most effectively in experiment FULL where the maximum value of the winter subtropical jet is 38 m s^{−1}. The position of the jet is slightly shifted poleward in comparison with the European Centre for Medium-Range Weather Forecasts (ECMWF) 40-yr Re-Analysis (ERA-40) data (Simmons and Gibson 2000). Furthermore, the model’s summer westerlies extend too high into the stratosphere. The dependence of the climatological zonal-mean zonal wind on the stationary wave forcing has been studied in detail by Becker and Schmitz (2001). There, it is shown that although orographic and thermal stationary wave forcings have opposite effects with regard to the Hadley cell, both forcings tend to reduce the jet maximum around 35°N.

To characterize the stationary waves, Fig. 3 displays the deviation from the zonal mean for the time-averaged geopotential at 300 hPa. In experiment AQUA (Fig. 3a), stationary wave components are negligible. In experiment TH (Fig. 3b), the stationary waves reach moderate values of up to 90 m. With orographic stationary wave forcing in the experiments ORO and FULL (Figs. 3c,d), strong stationary waves appear with maximum values of 230 and 280 m, respectively. Compared to the ERA-40 data, experiment FULL gives a realistic reproduction of the Northern Hemispheric winter state. On the other hand, the observed wintertime Southern Hemisphere with its weak stationary waves is rather similar to the experiment TH with reversed latitude.

The effect of the stationary wave forcing on the eddy kinetic energy is summarized in Table 1. On the longer time scales, that is, stationary, quasi-stationary, and low-frequency, the eddy kinetic energy increases with the strength of the stationary wave forcing in the order AQUA–TH–ORO–FULL. The high-frequency eddies show a more complex behavior. The orographic forcing shifts the whole spectrum to longer time scales thereby reducing the high-frequency eddy kinetic energy. This has also been observed in model studies of Yu and Hartmann (1995). However, the thermal stationary wave forcing concentrates the eddy kinetic energy in storm tracks close to the heating, which amplifies the high-frequency eddy kinetic energy. Because of this coupling the effective zonal-mean heating differs in midlatitudes for the different experiments. Yet these differences result only from the thermal stationary wave forcing and the nonlinear parameterization of the midlatitudinal condensational heating.

### c. Methods

#### 1) Empirical orthogonal functions

The leading variability patterns are determined in terms of empirical orthogonal functions (EOFs; Preisendorfer 1988). This method allows a straightforward comparison of variability patterns deduced from the different model experiments. The EOFs are calculated for the geopotential at 1000 hPa of the model’s Northern Hemisphere. Based on the definition of the Arctic Oscillation by Thompson and Wallace (1998), we use the equivalent geopotential height *Z*_{1000} = 8 m hPa^{−1} (*p _{s}* − 1000 hPa).

As will be shown later (section 3a), all model experiments exhibit an AM as the leading variability mode in the lower troposphere. For the analysis of the AM-related anomalies a composite analysis is performed by selecting 10% of all days with the highest and lowest values of the associated time series (principal components). If the principal component follows a Gaussian distribution, all values numerically larger than 1.28 times the standard deviation belong to either ensemble. The ensemble means are denominated by an overbar with the superscript AM± and measure ±1.64 standard deviations for a Gaussian distribution. These ensemble means define the high and low phase of the AM. The composite difference between the high and low phase is referred to as AM-composite difference or simply AM-related anomaly, marked with an overbar and the superscript AM.

#### 2) The zonal momentum budget

*u*and

*υ*denote the zonal and meridional velocity, Φ

*the geopotential of the orography,*

_{s}*ϕ*and

*λ*the latitude and longitude,

*p*the surface pressure,

_{s}*p*

_{0}a constant reference surface pressure,

*a*the earth’s radius, the angle brackets the vertical average, the square brackets the zonal mean, the star the deviation from the zonal mean, and

*F*the contribution from the surface drag. The vertical integration is carried out from

*p*

_{0}= 1000 to 100 hPa. The first term on the rhs of Eq. (2) is the convergence of the meridional eddy momentum flux; the second is the mountain torque. In the following, the surface drag is neglected, since it always damps the barotropic wind 〈[

*u*]〉. Yet the mountain torque is still retained, since it can either damp or amplify the wind. Daily and vertically integrated fields of the zonal-mean zonal wind, the eddy flux convergence, and the mountain torque are projected on the vertically integrated AM-composite difference of the zonal-mean zonal wind. This gives a time-dependent index for each term of Eq. (2), which describes the contribution to the AM-related zonal-wind anomaly. Here

*I*,

^{u}*I*, and

^{uυ}*I*denote the indices for the AM-related zonal-mean zonal wind, the eddy momentum flux convergence, and the mountain torque, respectively.

^{mt}Next, the variables of Eq. (2) are decomposed into a high-frequency, low-frequency, and quasi-stationary component: *X* = *X _{h}* +

*X*+

_{l}*X*. The high-frequency or synoptical component is determined with a Lanczos 10-day high-pass filter with 41 weights. We use a Lanczos 30-day low-pass filter with 61 weights for the quasi-stationary component, which comprises the planetary waves and especially the anomalous stationary waves of the AM. The low-frequency component contains the remaining time scales from 15 to 30 days. The meridional eddy momentum flux from Eq. (2) is now divided into a high-frequency part [

_{s}*u**

***

_{h}υ_{h}], a quasi-stationary part [

*u**

***

_{s}υ_{s}], and the remainder of the low-frequency and cross-frequency contributions [

*u**

***

_{h}υ_{l}+

*u**

***

_{l}υ_{h}+

*u**

***

_{h}υ_{s}+

*u**

***

_{s}υ_{h}+

*u**

***

_{s}υ_{l}+

*u**

***

_{l}υ_{s}+

*u**

***

_{l}υ_{l}]. These vertically integrated flux contributions are projected onto the vertically integrated AM-related zonal-mean zonal-wind anomaly, which yields the index

*I*for the high-frequency scales, the index

^{uυ}_{h}*I*for the quasi-stationary scales, and the index

^{uυ}_{s}*I*for the low-frequency time scales and cross-frequency contributions. Likewise, the index

^{uυ}_{r}*I*only includes the quasi-stationary fluctuation of the surface pressure. To evaluate the feedback processes in section 3b, time-lagged cross correlations (e.g., von Storch and Zwiers 1999, chapter 11.3) between the different time-filtered contributions of Eq. (2) and the AM-related zonal-wind fluctuation are used. Following Lorenz and Hartmann (2001, appendix B), we apply a significance test. For the construction of the Monte Carlo simulations, the artificial time series are assumed to have a memory of 6 days for the unfiltered and high-passed data. For the quasi-stationary data we determine the memory for each experiment as the first root of the autocorrelation of

^{mt}_{s}*I*or of (

^{uυ}_{s}*I*+

^{uυ}_{s}*I*), which yields values between 14 and 25 days. The applied significance level is 95%.

^{mt}_{s}#### 3) Linear model of the stationary waves

**y**comprises the spectral coefficients of vorticity, divergence, temperature, and surface pressure. Formally, we write

**y**= [

**y**] +

**y***, where [

**y**] and

**y*** represent all components with zonal wavenumber

*m*= 0 and

*m*≠ 0, respectively. The linearization is carried out about the average of [

**y**] over the high and low phase of the AM. The symbolic equation for the stationary waves in the AM phases isThe linear operator 𝗟

_{AM±}for the AM phases is determined tangentially via numerical differentiation of the dynamical core of the KMCM in spectral space (Valdes and Hoskins 1989). The stationary waves are forced by the terms in braces on the rhs of Eq. (3). All forcing fields are obtained from the data of the nonlinear KMCM experiments. The orographic forcing

*F**

_{oro}consists of the wave component of the model’s orography. Here

*F**

_{Q}contains the wave component of the diabatic heating

*Q*of Eq. (1), and the transient eddy forcing is TE. Finally, the stationary nonlinearity

^{AM±*}is the nonlinear self-interaction of the stationary waves, and it is by definition a nonlinear term;

^{AM±*}is calculated by subtracting the linear component 𝗟

_{AM±}

y

^{AM±*}from the total tendency calculated with the dynamical core of KMCM for the state vector averaged over the respective AM phase.

Linear models need additional linear damping to achieve quantitatively reasonable results (Grose and Hoskins 1979; Valdes and Hoskins 1989). For the present linear model, we apply 10 times the horizontal diffusion of the nonlinear model and a Rayleigh friction following the scheme of Nigam et al. (1986). The latter includes two components: a height-dependent friction coefficient for momentum and temperature starting from a value of 1 day^{−1} at the lowest level and decreasing to zero at about 800 hPa, and an extra coefficient for the momentum that depends on the background zonal-mean zonal wind. By construction, the forcing term SE includes the errors of the linear model due to linearization and changing the damping parameterizations.

*zonal–eddy coupling*in the linear analysis of DeWeaver and Nigam (2000b). The zonal–eddy coupling arises from the climatological stationary waves along with the different mean flow during the positive and negative AM phase.

Stationary wave forcing ^{c*} includes orography, diabatic heating, transient eddies, and the stationary nonlinearity. Especially, the latter ensures a realistic representation of the climatological stationary waves in the linear model of the AM phases. The second and third terms on the rhs of Eq. (7) describe the contributions of the anomalous forcing in the AM phases owing to anomalous diabatic heating and transient eddies. Further details on the linear model can be found in Körnich (2004).

## 3. Results

### a. Variability patterns

All the experiments show an AM-like pattern in their leading EOFs of the geopotential at 1000 hPa (Fig. 4). The displayed EOFs are weighted with the standard deviation of the respective principal component. Because the explained variances of the first two EOFs (Table 2) differ sufficiently, the patterns are not degenerate according to North et al. (1982), and we regard the first EOFs as the leading variability patterns. The low values of the explained variances in Table 2 result from fact that the variability of the geopotential is governed by the synoptical eddies. Using the low-passed geopotential, the first EOFs show almost the same patterns but with explained variances between 23% and 31%. As an estimate of the strength of the AM, the difference between the maximum and minimum value of the zonal-mean component of the pattern is compared for each experiment (Table 2). The value in experiment AQUA is 71.1 m. The orographic forcing alone in experiment ORO weakens the strength of the AM to 65.6 m, whereas the land–sea heating contrasts in the experiments TH and FULL lead to increased values of 92.5 and 104.6 m. The weak AM in experiment ORO is furthermore evident in the variability of the low-passed geopotential at 300 hPa. At 300 hPa, only the fourth EOF of experiment ORO displays an AM, with an explained variance of 8.4%, whereas the other experiments are dominated by a stronger AMs (see Körnich et al. 2003, their Fig. 1). At 1000 hPa, the AM of experiment FULL displays the strongest AM, and in addition, distinct centers of action over the Pacific and Atlantic Ocean (Fig. 4d). In this respect, the AM of experiment FULL is comparable to the observed northern AM (Thompson and Wallace 2000), while the observed southern AM with its strong zonal symmetry is more consistent with the AMs of the experiments AQUA and TH. Shortcomings of the model experiment FULL are the overemphasized zonal symmetry of the AM and the strong Pacific center of action.

To characterize the temporal behavior of the AM, we estimate the e^{−1} decorrelation time from the autocorrelation function of the first principal component (Table 2). This represents the memory of the AM. The longest decorrelation time is found in the experiment AQUA with a value of 23.2 days. If stationary wave forcing is taken into account, the decorrelation time decreases significantly. Especially in the experiments ORO and FULL with orographic forcing, the decorrelation time reaches low values of 7.8 and 14.8 days, while in experiment TH with the relatively weak forcing it is higher with 19.0 days.

We thus find that the strength of the AM and the memory of the simulated AM are smallest in experiment ORO, while the influence of the land–sea heating contrasts in the experiments TH and FULL leads to a stronger AM and a longer memory.

### b. The zonal momentum budget

Figure 5 shows the AM-composite differences of the zonal-mean zonal wind. In all experiments, the AM is related with a meridional dipole with extrema at about 35° and 55°N. Especially in high latitudes, the anomaly has a strong barotropic component up to approximately 200 hPa. In the troposphere, the smallest vertically averaged fluctuations appear in experiment ORO with about ±6 m s^{−1} (Fig. 5c), while experiment FULL shows the largest values of ±11 m s^{−1} (Fig. 5d). Additionally, the latter displays strong stratospheric fluctuations that are shifted poleward with height.

In the following, the time series analysis described in section 2 is applied in order to examine the maintenance of the AM. Since the analysis regards the zonal-mean angular momentum budget of the AM, the AM-related zonal-wind index *I ^{u}* is used. In each experiment, this index correlates highly (more than 0.95) with the first principal component of the zonal-mean zonal wind (zonal index). However, the correlation coefficient between the

*I*and time series of the AM is less (Table 2). Especially, the low correlation of 0.63 in experiment ORO points to the fact that the dynamics of the AM are only partly governed by the zonal index. For comparison reasons Table 2 includes also the

^{u}*e*

^{−1}decorrelation time of

*I*, where experiment ORO shows the lowest and experiment FULL the highest value.

^{u}Figure 6 shows the time-lagged cross correlation between the AM-related zonal-wind index *I ^{u}* and the sum of

*I*and

^{uυ}*I*. Negative time lag in the cross correlation means that the terms on the rhs of Eq. (2) lead the wind anomaly. For negative time lag, all experiments behave in a similar way. Thus, the sum of the meridional eddy momentum flux convergence and the mountain torque

^{mt}^{3}forces the AM-related wind anomaly in all experiments. Differences between the experiments appear for positive time lag, when the zonal-wind anomaly leads the forcing terms of eddy flux convergence and mountain torque. A positive correlation at positive time lag means that the wind anomaly induces a lagged forcing that maintains the wind anomaly. This corresponds to a positive feedback between the forcing terms and the wind anomaly (Lorenz and Hartmann 2001). All the experiments show statistically significant positive correlations over several days, from 6 to 30 days for AQUA, from 6 to 23 days for TH, from 9 to 11 days for ORO, and from 6 to 25 days for FULL all on a 95% level. The weakest correlations for positive time lag appear in experiment ORO (white dashed curve in Fig. 6). This agrees with the fact that this experiment, compared with the other experiment, shows the weakest AM with the lowest memory, both for the AM and the

*I*(Table 2).

^{u}Next, we consider the contribution of the different time scales to the maintenance of the AM-related zonal-mean zonal-wind anomaly. The cross covariances between the index of AM-related zonal-wind fluctuation *I ^{u}* and the time-filtered components of

*I*, as well as between

^{uυ}*I*and

^{u}*I*, are shown in Fig. 7. Each experiment shows a positive feedback between the high-frequency eddy flux convergence and the zonal-mean zonal-wind anomaly; that is, the high-frequency contribution of the eddy flux dominates the maintenance of the AM in all experiments (white solid curves in Fig. 7). Its positive values at positive time lags are significant on a 95% level from 6 to 30 days in all experiments, except in experiment ORO from 6 to 23 days. The contributions from the low-frequency and cross-frequency eddy fluxes (white dashed curves in Fig. 7) exhibit similar characteristics for negative time lag, and hence add to the forcing of the zonal-wind anomaly. However, with the predominantly negative values at positive time lag, these eddy fluxes tend to damp rather than maintain the zonal-wind anomaly.

^{mt}The important difference between the model experiments arises from the eddy fluxes with the quasi-stationary time scales (black dashed curves in Fig. 7). Since the persistence of the quasi-stationary eddy flux convergence is influenced by the low-pass filter, we determine their memory by the first root of the autocorrelation, which lies at 14, 20, and 25 days for TH, ORO, and FULL, respectively. The experiments AQUA and TH (Figs. 7a,b) indicate only small negative values for the cross covariance at positive time lag, so that the quasi-stationary eddy fluxes cannot feed back on the zonal-mean wind anomaly. Experiment ORO and FULL show significant positive values at positive time lags. However, the mountain torque (black dotted curves in Figs. 7c,d) counteracts the impact of the quasi-stationary eddy flux, and it shows negative cross covariance at positive time lag. Especially in a linear, inviscid stationary wave model, the mountain torque would exactly balance the stationary eddy flux convergence (Cook and Held 1992).

The black solid curves in Figs. 7c,d represent the sum of the quasi-stationary components of the eddy flux convergence and the mountain torque. The memory from the first root of the autocorrelation function yields now 14 and 18 days for ORO and FULL, respectively. In experiment ORO (Fig. 7c) the contribution of the mountain torque overcompensates for the eddy flux term, so that the total cross covariance of the quasi-stationary time scales is significantly negative until time lag 25 days. This means that a negative feedback between the quasi-stationary waves and the zonal-mean wind anomaly weakens the AM and its memory in experiment ORO. In contrast, the cross covariance of the quasi-stationary time scales in experiment FULL (black solid curve in Fig. 7d) remains weakly positive at positive time lags with significant values from time lag 22 to 25 days. Therefore, only experiment FULL yields a positive feedback between the quasi-stationary waves and the AM-related zonal-mean zonal-wind anomaly that compensates for the influence of the mountain torque. Although both terms, the eddy flux convergence and the mountain torque, are physically connected, the different behavior of the experiments arises from the feedback between the quasi-stationary waves and the zonal-mean flow, which is analyzed in the following.

### c. The feedback between the zonal-mean zonal wind and the anomalous stationary waves

Following DeWeaver and Nigam (2000b), the feedback between the zonal-mean zonal wind and the anomalous stationary waves during the AM phases consists of two steps. First, the zonal–eddy coupling induces anomalous stationary waves. Then, the anomalous eddy momentum flux convergence reinforces on the zonal-mean zonal-wind anomaly.

#### 1) The anomalous stationary waves

The linear stationary wave model is now applied to the AM phases in the experiments ORO and FULL. To describe the anomalous stationary waves of the AM phases, the geopotential at 300 hPa is used. For experiment FULL, Figs. 8a and 8b display the AM-related anomalous stationary waves from KMCM and the linear response to the zonal–eddy coupling [first term on the rhs of Eq. (7)]. The structures of both fields are in good agreement. Only the extended low over the North Pacific is not captured properly by the linear calculation. Additionally, the low over eastern Canada is too strong, while the highs over the Atlantic and Eurasia are underestimated. Nevertheless, the similarity indicates that the anomalous stationary waves result from the linear wave propagation in the zonal-mean background of the respective AM phase. The anomalous forcing of the AM phases [second and third terms on the rhs of Eq. (7)] contributes only weakly to the anomalous stationary waves slightly compensating for the zonal–eddy coupling in high latitudes (not shown).

Figures 8c and 8d show the AM-related anomalous stationary waves of KMCM and the zonal–eddy coupling term for experiment ORO. The anomalous stationary waves consist of a wave train with high amplitudes from the Atlantic to Siberia (Fig. 8c). These waves are not reproduced by the linear zonal–eddy coupling (Fig. 8d), the effect of which is negligible in experiment ORO. Since even the anomalous forcing in the AM phases [second and third terms of Eq. (7)] fails to describe the anomalous stationary waves (not shown), these waves are determined by their nonlinear self-interaction.

#### 2) Feedback on the zonal-mean zonal wind

*G*of the anomalous meridional eddy momentum flux convergence:The superscript c denotes the climatological mean. The anomalous stationary waves are calculated either linearly from the zonal–eddy coupling term of Eq. (7) or as an ensemble mean from the KMCM data.

_{a}The vertically integrated term *G _{a}* with the anomalous stationary waves computed from the zonal–eddy coupling is shown in Figs. 9a–c for the experiments TH, ORO, and FULL (white dashed curves). The black dotted curves display the corresponding AM-related zonal-mean zonal-wind anomalies. Only for experiment FULL does

*G*due to the zonal–eddy coupling reinforce the anomalous westerlies around 55°N (Fig. 9c). Thus, experiment FULL exhibits a positive feedback between the zonal-mean zonal wind and the anomalous stationary waves of the AM. In the experiments TH and ORO (Figs. 9a,b), the zonal–eddy coupling induces only a weak eddy flux convergence that does not coincide with the wind anomaly and therefore describes no feedback.

_{a}To compare the linear model results with the KMCM data, Figs. 9a–c display additionally the vertically integrated *G _{a}* with the AM-related anomalous stationary waves taken from KMCM data (white solid curves) as well as the respective mountain torque (white dotted curves). Whereas both terms tend to cancel each other and yield only a weak projection on the AM in experiment ORO (Fig. 9b), we find indeed a reinforcement of the anomalous westerlies in experiment FULL (Fig. 9c). Furthermore, in this experiment, the quasi-linear stationary component

*G*contributes significantly to the total anomalous momentum flux convergence (black solid curves in Figs. 9a–c). In experiment ORO (Fig. 9b), the contributions of the anomalous stationary eddies cannot be maintained by the zonal–eddy coupling, so that the damping impact of the mountain torque on the AM dominates for the positive time lag in the zonal momentum budget (Fig. 7c). For experiment TH (Fig. 9a) the anomalous stationary wave contribution is negligible in the entire hemisphere.

_{a}From the comparison of the total anomalous eddy flux convergences with their stationary component follows that the AMs of all experiments are strongly maintained by anomalous transient eddies, and particularly by the high-frequency or synoptical contribution, as was shown by Körnich et al. (2003). On the other hand, only the combination of orography and land–sea heating contrasts leads to a positive feedback between the zonal-mean zonal wind and the anomalous stationary waves of the AM phases, which, in turn, result from the zonal–eddy coupling. In other words, zonal–eddy coupling induces anomalous stationary waves that feed back on the zonal-mean zonal-wind anomaly by their quasi-linear interaction with the climatological stationary waves.

The different behavior of *G _{a}* in the experiments ORO and FULL can result from either the different zonal-mean background flows or from the different stationary wave forcings. To separate these effects, two additional calculations are carried out, where both the climatological and the anomalous stationary waves in Eq. (8) are determined with the linear model.

In the first calculation Lin1, the climatological wave forcing of experiment FULL is used along with the climatological zonal-mean flow of experiment ORO to compute the climatological stationary waves. The AM-related anomalous stationary waves are obtained by disturbing the zonal-mean flow of experiment ORO with the zonal-mean anomalies of experiment FULL, while the wave forcing from experiment FULL is retained. With this experiment, the effect of the “right” stationary wave forcing of experiment FULL in presence of the “wrong” zonal-mean climate of experiment ORO is demonstrated. The resulting *G _{a}* of experiment Lin1 (solid curve in Fig. 9d) yields a strong positive signal at 60°N, which indicates a feedback with the AM.

In the second calculation Lin2, the climatological stationary waves are computed for the zonal-mean climate of experiment FULL with the climatological stationary wave forcing of experiment ORO. The anomalous stationary waves are determined by applying the wave forcing of experiment ORO on the zonal-mean flows of the AM phases from experiment FULL. This experiment includes the “wrong” stationary wave forcing but the “right” zonal-mean climate. The *G _{a}* of experiment Lin2 (dashed curve in Fig. 9d) gives a significantly reduced dipole pattern that still projects on the AM-related zonal-wind anomaly. However, the comparison of these additional calculations with experiment FULL (Fig. 9c) indicates that it is the combined stationary wave forcing by orography and land–sea heating contrasts that mainly causes the feedback process, while the zonal-mean climate is of secondary importance.

## 4. Discussion

The role of baroclinic and stationary waves for the maintenance of the AM has been examined using GCM experiments with different stationary wave forcing. With no or weak stationary wave forcing in the experiments AQUA or TH, the maintenance of the AM is dominated by the high-frequency eddies with periods less than 10 days, and the influence of anomalous stationary waves is negligible. The rather zonally symmetrical AMs in these experiments resemble therefore the observed southern AM (Limpasuvan and Hartmann 2000; Lorenz and Hartmann 2001). Only with stationary wave forcing by orography along with land–sea heating contrasts (experiment FULL) does the model simulate a strong AM that is comparable to the observed northern AM. In the simulated AM the zonal symmetry and the center of action over the Pacific are somewhat too strong. Nevertheless, the linear stationary wave propagation in the different zonal-mean background flows of the AM phases can explain the anomalous stationary waves, especially in the North Atlantic region. This result corresponds to the zonal–eddy coupling analysis of the observed NAO by DeWeaver and Nigam (2000b). Furthermore, the convergence of the meridional eddy momentum flux due to the zonal–eddy coupling calculated with the linear model and with the KMCM data accords with the respective analysis of the observed Northern Hemisphere (Limpasuvan and Hartmann 2000; DeWeaver and Nigam 2000a, b). Therefore, our mechanistic model experiment with orography and land–sea heating contrasts reproduces the main dynamic process of the northern AM.

The model experiments suggest how to interpret the difference between the observed northern and southern AM. Even though all of our experiments share the feedback between the high-frequency eddies and the anomalous westerlies of the AM phases, we find that the stationary wave forcing is not merely inducing a longitudinal dependence of the AM, but that it does influence the zonally symmetric component of the AM as well. The strong influence of stationary wave forcing on the zonal-mean component of the AM becomes apparent when comparing the experiments ORO and FULL, which both include the same orography. In experiment ORO with no land–sea heating contrasts, the feedback due to zonal–eddy coupling is absent, and the zonally symmetric component of the AM is weakened by the mountain torque, while in experiment FULL zonal–eddy coupling enhances the zonal-mean zonal-wind oscillation. Thus, we conclude that the feedback between the anomalous stationary waves and the zonal-mean zonal wind is necessary for the simulation of the observed northern AM. This process distinguishes the northern AM from the southern AM.

Besides the anomalous stationary waves, the transient eddies, especially their high-frequency baroclinic components, play a crucial role in the maintenance of the AM. A possible interpretation of the feedback between synoptical waves and the zonal index has been proposed by Robinson (2000). Each of our model experiments exhibits a positive feedback between the high-frequency eddy flux and the zonal-mean zonal-wind anomaly in the cross covariances. The high-frequency eddy kinetic energy (Table 1) and thereby the strength of this feedback is also influenced by the stationary wave forcing. While the orographic forcing reduces the high-frequency eddies, it raises the low-frequency eddies, which do not feed back positively on the zonal-wind anomalies (Yu and Hartmann 1995). On the other hand, the thermal stationary wave forcing increases the high-frequency eddy kinetic energy by forming storm tracks in the regions of heating. So the changes in the strength and persistence of the AMs in the experiments seem to be influenced by the changes in the eddy kinetic energy. Indeed, the experiment TH shows both stronger high-frequency eddy kinetic energy and a stronger AM than experiment AQUA (Tables 1 and 2). The weak wave component of the AM in experiment TH is consistent with the fact that the stronger transient eddies damp the stationary waves in a linear model (Nigam et al. 1986). However, the stronger AM in experiment FULL compared to experiment TH is accompanied by weaker high-frequency eddy kinetic energy. It should be noted that the effect of the high-frequency eddy flux on the zonal-mean anomaly is slightly underestimated in experiment FULL (Fig. 7d) in comparison with observational data of the Northern Hemisphere (Lorenz and Hartmann 2003, their Fig. 8a).

Additionally, the transient eddies influence the anomalous stationary waves in two ways. There is an indirect effect through the enhancement of the zonal-mean zonal-wind anomaly by the transient eddies. This zonal-mean anomaly is maintaining the anomalous stationary waves of the AM phases by the zonal–eddy coupling. This indirect effect seems to be consistent with Feldstein (2003) and Lorenz and Hartmann (2003), who accentuate the role of the transient eddies for the forcing and the maintenance of the observed NAO and NAM. However, when the direct effect of the transient eddy forcing in Eq. (3) on the anomalous stationary waves is computed with the linear model for our experiment FULL, we find the opposite effect, that is, a weakening of the anomalous stationary waves (cf. Fig. 10 to Fig. 8a). Since the transient eddies both affect the zonal-mean state and the stationary waves, it seems very difficult to separate both effects.

Lorenz and Hartmann (2003) demonstrate that the waves on the different time scales play different roles for the observed northern AM. High-frequency and quasi-stationary waves yield a positive feedback, while the low-frequency and cross-frequency contributions give a negative one. This negative feedback results from the external character of the low-frequency Rossby waves. We transfer these results to our GCM experiments in order to understand the effect of the combined stationary wave forcing of orography and land–sea heating contrasts on the AM (Fig. 9d).

Figure 11 shows the climatological stationary waves of the experiments as a function of longitude and pressure. Experiment FULL agrees well with observations (e.g., Wallace 1983), featuring a westward phase shift of the stationary waves with height, while the orographic and thermal forcing experiments (TH and ORO) are comparable with the corresponding model results of Held (1983). Larger horizontal scales dominate in experiment TH than in experiment ORO, where furthermore the stationary waves possess the structure of external Rossby waves. Held (1983) interprets these characteristics of the stationary waves as follows: Even if the orographic and the thermal forcing had similar horizontal scales, the orography forces stationary waves of smaller scales than the heating, because the equivalent topography of the heating is proportional to the reciprocal of the horizontal wavenumber, which means more weight on the larger scales. In addition, the orography is rather localized (Fig. 1a), so that the far field model response consists of external Rossby waves. In contrast, the time-mean heating (Fig. 1c) exhibits sufficient large scales in comparison with the external Rossby waves, so that the low-level thermal forcing produces a westward phase shift of the stationary waves with height for the experiments TH and FULL.

These considerations apply as well to the time scale of the AM composites and the anomalous stationary waves of the experiments (Fig. 12). Therein lies the importance of the thermal stationary wave forcing for the AM in experiment FULL. In contrast to the external Rossby waves for the orography alone (Fig. 12a), the thermal forcing produces again the stronger westward phase shift with height (Fig. 12b), which enables a wave-mean flow interaction and thus the positive feedback with the AM in experiment FULL. It would be interesting to see whether this relative importance of orographic and thermal stationary wave forcing for the northern AM can be used to understand observed changes of the variability patterns.

Finally, stratospheric variability seems to be related to the tropospheric AM, as the AM-related zonal-mean wind anomaly in experiment FULL indicates (Fig. 5d). One mechanism of this vertical coupling can result from the changed planetary wave propagation in the AM phases, which influences the upward Eliassen–Palm flux from the troposphere into the stratosphere. This anomalous Eliassen–Palm flux is evident in our model experiment and in observational data (Körnich et al. 2003; Hartmann et al. 2000). We will address this problem in a companion study.

## 5. Conclusions

The influence of stationary wave forcing by orography and land–sea heating contrasts on the AM has been investigated using a simple global circulation model with perpetual January conditions. Experiments with different stationary wave forcings all show an AM in the lower troposphere, but with different strengths of the zonally symmetric component and different longitudinal dependencies. Time series analysis of the vertically integrated zonal momentum budget confirms that the anomalous high-frequency eddies with periods less than 10 days always force and maintain the AM. For the experiments without orography, the high-frequency part of the meridional eddy momentum flux clearly dominates the zonal-mean zonal-wind budget during the AM phases. This makes the AMs of these experiments comparable to the observed southern AM (Limpasuvan and Hartmann 2000). The stationary wave forcing changes the transient eddy kinetic energy on all time scales, where orographic forcing weakens and thermal forcing strengthens the high-frequency transients. By this, the stationary wave forcing can affect the strength of the AM.

The influence of the AM-related stationary wave anomalies differs substantially among the experiments. The momentum budget analysis shows that the mountain torque generally weakens the barotropic zonal-mean zonal-wind anomaly. This effect can be compensated for by the feedback of anomalous stationary waves generated by the zonal–eddy coupling (DeWeaver and Nigam 2000b). When this feedback is absent, as in the experiment with orographic stationary wave forcing alone, the AM and its memory are weakened. Only for the combined stationary wave forcing by orography and land sea heating contrasts does the feedback of the anomalous stationary waves lead to a strong AM with a superposed longitudinal structure. The different behavior of the model experiments results from the fact that only the diabatic heating changes the character of the stationary waves from external Rossby waves for orography alone to long vertically propagating waves that interact with the zonal-mean flow and induce the feedback process. Relating our model results to the observations, a realistic simulation of the northern AM is achieved only with this process. Thus, the model experiments enlighten the difference between the observed northern and southern AM as well as the relative importance of orography and land–sea heating contrasts for the dynamics of the AM.

The valuable comments of D. L. Hartmann and two anonymous reviewers are gratefully acknowledged. The authors are thankful to the ECMWF for providing the ERA-40 data.

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Description of model experiments (see text for explanation of forcing) and their climatological distribution of the eddy kinetic energy on the different time scales (stationary, high frequency with periods shorter than 10 days, quasi stationary with periods longer than 30 days, and low frequency for the remaining time scales and cross-frequency contributions).

For the EOFs of Fig. 4, this table shows the explained variance of the first two EOFs with the second in parentheses, the strength of the AM defined by the difference between the maximum and the minimum of the zonally averaged weighted first EOF, the *e*^{−1} decorrelation times in days for the first principal component and for the AM-related zonal-wind fluctuation *I ^{u}* [section 2c(2)], and the correlation coefficient

*r*between the first principal component and

*I*.

^{u}^{1}

We use the term “anomaly” as the change owing to the high or low phase of a variability pattern in respect to the climate mean state.

^{2}

In Körnich et al. (2003), the experiments ORO and TH are referred to as WO and QCM.

^{3}

The mountain torque is zero in the experiments AQUA and TH.