a. Model

In Part I, we proposed a dynamical nonlocal closure to describe the two-way SELF feedback using an anomaly modeling approach by considering a stochastic basic flow. This basic flow consists of the climatological mean basic flow, denoted as ψ_c(x, y), and a stochastic synoptic eddy flow, denoted as Ψ′_c. The latter is expressed as

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Here c.c. stands for complex conjugate. Each pair of E_n(x, y)e^iω_nt represents an independent propagating pattern with angular frequency ω_n, characterized by three other parameters: a fractional variance to the stochastic eddy flow, σ_n, and complex time coefficient ξ_n(t), which follows an independent Gaussian process with an e-folding time τ_n. The entire set (E_n, σ_n, τ_n, ω_n; n = 1, N_c), which captures the observed climatological characteristics of the synoptic eddy flow, is determined from daily 500-hPa streamfunction data obtained from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis for the cold season (April–November for NH and May–October for SH) during 1979–95. Interested readers should refer to Part I for details.

Based on the results obtained in Eq. (11) of Part I, the linear barotropic model for anomalous time–mean flow can be expressed as

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Here ψ^a denotes the streamfunction field of the time-mean flow anomalies, or loosely the low-frequency anomaly, whereas Q^a represents the external forcing. For any field, the linear operator L is defined as

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where J is the Jacobian operator, f is the Coriolis parameter, and ψ^c is the observed climatological mean field of the streamfunction. The linear friction coefficient r is set at 1 (10 days)⁻¹. The only new term in Eq. (3), which makes it different from the traditional linear barotropic model such as that of S83, is the eddy feedback term L_fψ^a. Utilizing Eq. (10) of Part I and the so-called quasi-equilibrium approximation, we can express this extra term as follows:

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Here Ẽ_n is the complex conjugate of E_n. Each component ψ̂_n corresponds to a complex pattern that is related to the covariance between anomalous and basic stochastic eddy flows. Eliminating ψ̂_n by inverting the equations in the second line of Eq. (4), we obtain the expression for SELF feedback operator L_f as shown in Eq. (13) of Part I. Equations (2)–(4) thus form a linear barotropic model for modeling low-frequency flow anomalies with the SELF interaction.

b. Singular value decomposition analysis

It has been shown that the steady response to spatially white forcing in the streamfunction field is dominated by the left singular vectors of the linear dynamical L + L_f with smallest singular values (cf. Branstator 1990; Navarra 1993; Itoh and Kimoto 1999; Watanabe and Jin 2004). Thus, to investigate the origin of the lowfrequency modes, we perform singular vector analysis to the linear operator of the system L + L_f by means of singular value decomposition (SVD). In this section, we adopt a spectral model version of the system (2)–(4) with T42 resolution for all numerical calculations.

The leading singular modes of the dynamical system (2) are shown in Figs. 1 and 2, respectively. The spectra of the singular values suggest that there is one distinct leading singular mode in each hemisphere. Although the magnitudes of the leading singular value depend on the linear damping rates, the patterns of the singular modes are robust to the choice of this relatively arbitrary parameter of the model.

The leading singular mode in the SH has a dipole component of the zonal flow and it resembles the observed pattern of the AAO as seen in Fig. 3a of Part I. In addition to the dipole component of the zonal flow, the observed streamfunction pattern associated with the AAO has some pronounced stationary wave features dominated by wavenumbers 2 and 3, as marked by two negative centers in the polar region and three positive centers in the midlatitude belt (Fig. 3a of Part I). Remarkably, the simulated AAO-like leading singular mode presented in Fig. 1a also has similar features, with arguably two negative centers in the polar cap and three positive centers in the midlatitude belt, although the centers in the solution here are somewhat shifted and weaker when compared to observations.

Similarly, the leading singular mode (Fig. 2a) in the NH resembles the AO/NAO, which also accompanies a zonal mean flow anomaly (Fig. 2b) in the leading mode. The dipole structure in the streamfunction becomes locally pronounced over the Atlantic sector, akin to the NAO. When it is compared with the observation as shown in Fig. 3b of Part I, the simulated center in the Atlantic is relatively weak and shifted toward the eastern Atlantic. Nevertheless, the dipole in the zonal flow and local enhancement over the Atlantic sector of the observed AO/NAO are qualitatively captured by this leading mode.

To isolate the role of the SELF feedback, we examine the leading modes of the dynamical system without L_f . In this case, the linear model described in Eq. (2) returns to its traditional form (e.g., S83 model), except for a different choice of the basic state. Without L_f , the leading singular modes of the barotropic model linearized with respect to the climatological mean flow of NH or SH winter circulation, as shown in Figs. 3 and 4, bear little resemblance to the AO or AAO. The second mode in the NH has a dipole structure in the zonal mean wind, but the circulation pattern is still drastically different from the observed AO (not shown). The above results suggest that in the barotropic framework (1) the SELF feedback is responsible for generating the AO- and AAO-like leading singular vectors as the leading low-frequency modes of the system. Although the leading singular vector of baroclinic models without SELF feedback may exhibit some AO likeness as well (Kimoto et al. 2001; Watanabe and Jin 2004), we found in Pan et al. (2006, hereafter Part III) that the SELF feedback play an important role in the generation AAO- and AO-like modes in the baroclinic framework (Part III).

In the system (2), the AAO- and AO-like modes have two major energy sources in the basic flow, namely, the climatological mean flow and the basic synoptic eddy flow. Both of them are also the sources for zonal asymmetries seen in the AAO- and AO-like modes. The climatological mean flow is not zonally uniform owing to the presence of stationary waves. The storm tracks are not zonally uniform, even in the SH, whereas there are so-called Pacific and Atlantic storm tracks in the NH. These two zonal asymmetries in the basic stochastic flow are ultimately due to topography and land–sea contrasts underlying the atmosphere; they are dynamically related, as can be inferred from the collocations of the jet streams and storm tracks in the basic state as seen in Fig. 1 of Part I, for example. In our linearized framework, these two parts of basic flow and the asymmetries in this flow are prescribed. Thus, we can examine separately the impacts of these asymmetries on the leading low-frequency modes.

Because we have already examined the case without L_f , that is, without the climatological storm track activity in the basic state thus without the energy source from the basic synoptic eddy flow, we now study the case without the effect of stationary waves of the basic flow in the linear operator L. We do so by considering only the zonal mean part of the climatological flow ψ^c for the linear advection terms J(ψ^c, Δψ^a) + J(ψ^a, Δψ^c + f ) in the linear operator L of Eq. (2). In this way, we have suppressed the S83 mechanism of nonzonal flow instability for low-frequency modes. All asymmetric features of the leading modes in this case can be completely attributed to the zonal asymmetry associated with L_f . For the SH leading mode, as shown in Fig. 5, there is a pronounced local enhancement between the Australia and Antarctic sector, downstream of the location of the maximum storm track intensity. This localized dipole feature is present in the AAO-like mode and can now be attributed to the zonal variation in the SELF feedback. In the NH case, we again find that the leading mode (Fig. 6) is an AO/NAO-like mode and the solution is very similar to that shown in Fig. 2, indicating that the local features in the northern Atlantic thus are also attributable to the SELF feedback. The modest differences in the patterns of the singular modes shown in Figs. 1 and 2 and Figs. 5 and 6 indicate that the SELF interaction controls the main features of the leading modes of Eq. (1).

In addition to the leading AAO- and AO-like modes, other singular modes also reveal some interesting features. For instance, the third mode of system (1) under the same background conditions as the case shown in Fig. 6 has a significant projection onto the observed PNA pattern (Fig. 7). The third singular mode under the background conditions as in the case shown in Fig. 2 also contains the PNA-like structure in the Pacific–North America sector. Thus, the SELF feedback plays an important role in the generation of PNA-like teleconnection pattern, as also indicated in an example in Part I.

c. Eigenanalysis

The SVD analysis to the linear dynamical system implies that spatially random forcing will preferentially excite the most singular, or near-neutral, modes of the system. The leading singular modes are often similar to the least damped eigenmodes of the linear dynamical operator L + L_f , particularly when we focus on the low-frequency modes of zero or near-zero frequencies. In other words, the least-damped eigenmodes of L + L_f also contain AO-, PNA-, and AAO-like features. For brevity, we will not show the patterns of leading eigenmodes of the linear dynamical system (1). Instead, we examine how the leading eigenvalues change between the cases without and with the SELF feedback. This is done by considering an artificial nondimensional parameter μ, varying from zero to one and then tracing the numerically changes in eigenvalues of L + μL_f as the function of μ.

The decay rates of the leading low-frequency modes systematically reduce as μ increases (Fig. 8a) from zero to one. For the least damped modes in both NH and SH (Fig. 8b), the decay rate is reduced from about 8–9 days⁻¹ to 11–12 days⁻¹, indicating that the SELF feedback serves as a significant positive feedback corresponding to a positive growth rate of about 30 days⁻¹. This positive feedback is similarly found for the leading modes in both SH and NH. We also notice that eigenmodes having smaller spatial scales tend to be more damped when the SELF feedback is included. Although the positive SELF feedback is not strong enough to overcome frictional damping in the model, its scale-dependence provides a natural selection favoring the planetary-scale low-frequency modes The scale-selectiveness of the positive SELF feedback will be further demonstrated in the next two sections with an idealized model and an analytical model.

a. Analytical model

To extract essence, we make the following simplifications. We consider a β-plane channel version of the barotropic vorticity equation in Cartesian coordinates. The climatological flow ψ^c is now given as

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and u^c is a constant zonal flow. The stochastic basic eddy flow for the climatological storm track activity is also drastically simplified by considering only a single propagating wave:

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where Ψ′_c is an idealized representation of the storm track with zonally uniform variance, G(y) is the meridional structure of the transient wave, k is the zonal wavenumber, and c is the propagation speed. As in Eq. (1), we assume that ξ(t) represents a complex normalized red noise process. Therefore, in this simple case, the set (E_n, σ_n, τ_n, ω_n; n = 1, N_c) for characterizing the background synoptic eddy activity is reduced to

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where τ is the e-folding time of the autocorrelation function from ξ(t).

Substituting Eqs. (5) and (7) into Eqs. (2)–(4), we can obtain a linear equation for the low-frequency flow anomaly with respect to this simple stochastic basic flow ψ^c + Ψ′_c. For further simplicity, we focus on the zonal mean part of the low-frequency flow anomaly ψ^a. Thus, we only consider the zonal mean part of Eq. (1), which can be rewritten as

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where u^a = −∂ψ^a/∂y, denoting the low-frequency zonal-mean flow anomaly. In the above equation for u^a, the last term, obtained by integrating with respect to y for the zonal mean of Eq. (4), represents the anomalous convergence of the ensemble mean of eddy momentum flux. In Eq. (8), is the complex conjugate of ψ̂(y, t); and ψ̂(y, t), which is governed by the following equation:

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This is a simplified version of Eq. (4) by setting ψ̂₁ = ψ̂(y, t)e^ikx and r^τ = 1/τ. Equations (8)–(9) describe the zonal wave interaction with respect to the stochastic basic flow ψ^c + Ψ′_c. The zonal flow anomaly u^a induces an anomalous stochastic Rossby wave through anomalous advections of the basic stochastic eddy flow Ψ′_c. Such an anomalous Rossby wave is related to ξ(t), and their covariance field ψ̂₁ thus follows the variations in u^a. The amplitude of ψ̂₁, ψ̂(y, t), is described by Eq. (9). This anomalous Rossby wave leads to an anomaly in the ensemble mean of eddy momentum flux, which feeds back to the zonal flow anomaly u^a, as described by the last term in Eq. (8).

b. Tilted trough mechanism and SELF feedback

Eliminating ψ̂ in the Eq. (8), one obtains the SELF closure for u^a. The closure operator is still rather complicated, and we will assess the effect of the SELF feedback by directly solving Eqs. (8)–(9). We consider the following approximations:

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Here, l² and l²_c represents typical meridional scales for anomalous zonal mean flow and the stochastic synoptic wave, respectively.

When l² ≫ l²_c + k², the meridional scale of the zonal flow anomaly is much smaller than the scale of the synoptic eddies in the basic flow. In this case, Eq. (9) can be approximated as

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The anomalous eddy flow is mainly due to the vorticity advection of the anomalous zonal flow by the basic synoptic eddy flow.

If we only keep the dominating term of the anomalous eddy momentum flux in Eq. (8), we get

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where the eddy viscosity coefficient is

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If the eddies are nearly isotropic so that l_c ∼ k, then 2|kG|² is a measure of eddy kinetic energy and the eddy diffusion coefficient is thus proportional to the total eddy kinetic energy in the basic state. Here κ depends also on k(u^c − c)/(r + r^τ), the ratio of the advection (propagation) and decay time scales of the synoptic eddies. Normally this ratio is larger than unity. If c = u_c − β/(l²_c + k²) and k(u^c − c)/(r + r^τ) ≫ 1, then κ = 8|kG|²l²_c(r + r^τ)/β². Setting the linear friction time scale to be about 10 days, the e-folding time of eddy lifetime to be about 3–4 days, 2|kG|² to be O(100 m² s⁻²), and l_c to be O(10⁻⁶ m), we find that κ ≈ 10⁶ m² s⁻¹. This is a significant eddy viscous damping for zonal flow anomalies with relatively small spatial scales (l² ≫ l²_c + k²).

The schematic diagram in Fig. 10 illustrates how the SELF feedback tends to damp the zonal flow anomalies of relatively small meridional scales. Let us start with an initial zonal flow anomaly with an anticyclonic shear or a negative vorticity. Since the meridional scale of the zonal flow is smaller than that of the stochastic wave in the basic flow, the anomalous eddy flow is mainly owing to the meridional advection of vorticity in the anomalous zonal flow by the stochastic wave in the basic flow. The anomalous eddy flow thus has a similar meridional scale to that of the zonal flow anomalies. Superimposed on the stochastic wave in the basic flow, it results in systematic NW–SE tilted troughs and ridges, as shown in the third column of Fig. 10. The ensemble and zonal mean of these tilted transient waves induce a net ensemble-mean and zonal mean northward eddy-momentum transport in the perturbed region, which act to reduce the initial vorticity in the zonal flow anomaly. Thus amidst the idealized basic synoptic eddy field representing the climatological storm track, the zonal flow modes with the meridional scales much smaller than the meridional scale of synoptic eddies are damped by the eddy viscosity due to this negative SELF feedback.

Now consider another case in a different flow regime where the zonal-mean flow anomalies have relatively large meridional scales; that is, l² ≪ l²_c + k². In this case, we have the following approximation from Eq. (9),

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Now the eddy flow is mainly the result of the zonal advection of the vorticity in the stochastic basic wave by the anomalous zonal-mean flow. The two terms for the anomalous eddy momentum fluxes in the zonal mean flow Eq. (8) are of the same order. We now get the following approximate equation:

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Here we have assumed c = u^c − β/(l²_c + k²). If we again consider that 2|kG|² is O(100 m² s⁻²) and the eddy e-folding time is O(3–4 days), we obtain κ₋ ∼ −10⁷ m² s⁻¹. This is a significant negative viscosity for relatively large-scale zonal flow anomalies. Though l² is small for zonal-mean flow anomalies in this flow regime, the destabilization for the zonal-mean flow anomalies due to this positive SELF feedback still can be significant.

Conceptually, this positive feedback can be also understood by the tilted trough mechanism. As illustrated in Fig. 11, an initial sheared zonal flow anomaly systematically alters the synoptic waves in the climatological storm track through the advective process. This leads to a systematic NE–SW tilting in eddies. The systematic tilted synoptic eddies in this case further results in an enhancement of the net ensemble- and zonal-mean momentum fluxes. As the result, the altered storm track has a net effect to reinforce the initial zonal-mean flow anomaly, constituting a positive SELF feedback. This kind of mechanism was also found to be operating in the stationary wave and zonal flow feedback (Kimoto et al. 2001).

c. Scale selection of SELF feedback

The above qualitative analysis clearly indicates that the SELF feedback is scale-dependent. When l² ≫ l²_c + k², the negative SELF feedback suppresses the formation of small-scale zonal flow anomalies through a viscous-type damping. When l² ≪ l²_c + k², the SELF feedback results in a positive feedback for zonal flow modes. Somewhere in the regime near l² ≈ l²_c + k², the SELF feedback changes sign. Thus the zonal flow mode has a positive feedback in the regime 0 < l² ≤ l²_c + k². This scale dependence of the SELF feedback suggests that there is a remarkable internal dynamical selection that favors a particular meridional scale of the zonal-mean flow modes that emerge from the stochastic basic flow.

We can derive an analytical expression for this scale-dependence of the growth rate for the zonal-mean flow modes by using the following approximation method. Assuming G ∼ σ sinl_cy, with σ a constant measuring the intensity of the variance in the synoptic eddy flow, and denoting the imaginary part of ψ̂ as Y, and further letting

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we can rewrite Eqs. (8)–(9) as

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We consider an approximate form of the solution for the above equations:

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and take a truncation ∂²Y/∂y² ∼ −(l² + l²_c)Y, which ignores some terms having little projection on the structures of Y or u^a. Then, we obtain an approximate equation for zonal modes as follows:

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Here the feedback factor is

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where we have set 2|G|² ≈ σ², which is the variance measure of the zonally uniform storm track in terms of a streamfunction. It becomes apparent that the sign of the SELF feedback depends on the sign of k² + l²_c − l². It is also clear that the SELF feedback is proportional to the strength of the storm track variance σ²; that is, the stronger the storm track activity, the stronger the SELF feedback. For the zonal modes with meridional scales such that l² < k² + l²_c, the positive SELF feedback tends to reduce the linear damping by a significant fraction, which is also seen from the eigenanalysis in section 2c. When σ² is very large, the positive feedback may overcome the linear frictional damping; hence the zonal model becomes linearly unstable. In the nonlinear case, this instability leads to multiple stochastic equilibria (Farrell and Ioannou 2003; Kravtsov et al. 2003, 2005). However, numerical results shown in section 2c suggest that the positive SELF feedback based on the observed levels of the storm track activity is modest. The linear system (2) is stable unless the linear friction is set to be unrealistically weak.

From Eq. (17), the least-damped mode of the zonal flow has a meridional scale about l_m ≈ (k²_c + l²_c)/2. With a reasonable choice of parameters, the most favored zonal flow mode is around l = 2, which corresponds to a dipole structure in the zonal flow. The dipole structure of the zonal flow, u^a = U_l sin2y, qualitatively resembles the observed structure in the zonal components of the AO and AAO patterns in Figs. 1 and 2. A similar equation to (16) for the AAO index or annular mode index of the SH was empirically derived by Lorenz and Hartmann (2001). However, our analytical model provides a mechanism for the observed meridional structure of the zonal mean flow anomalies associated with AAO and AO modes.

Moreover, the associated anomalous variance field of synoptic eddies also has a dipole structure which is proportional to the meridional structure of GY or |G²|u^a. The dipole-structured low-frequency zonal flow anomalies associated with the AO and AAO shifts the position of jet streams. At the same time, the associated anomalous transient eddy activity shifts the position of maximum variance in the storm track as well. Our analysis suggests that the atmospheric internal dynamics allows anomalous global-scale circulations of AO- and AAO-like structures to organize the synoptic eddies in the turbulent background atmospheric circulation in such a way as to gain selective reinforcement. This dynamic self-organization of the AO and AAO mode is likely essential for them being the leading modes of variability in the extratropical atmospheric circulation.

The analytical result in Eq. (17) may not be very accurate when l is close to l_c. Thus, we solve the eigenvalue numerically using Eqs. (8)–(9). The numerical and analytical results agree well in terms of feedback factors as shown in Fig. 12, suggesting that the approximate analytical solution is qualitatively correct. The results suggest that modes with meridional wavenumber 2, namely the modes of dipole structure, is favored, which is also consistent with numerical result in section 3 for the case under the basic flow with the zonally uniform storm track in section 3.

It is difficulty to obtain analytical results for background conditions with longitudinally confined synoptic activity as in the idealized cases section 3. Numerical results in section 3 suggest that the zonal extent of the enhanced synoptic activity in the background conditions play a crucial role in controlling the local features of the AO and AAO. The analytical results in this section show clearly that the key feature of the AO and AAO, namely their dipole structures, is the consequence of natural selection by the SELF feedback.