## 1. Introduction

The atmospheric teleconnection pattern is an important low-frequency phenomenon of climate variability. Although there are considerable studies on the causes, variations, and influences of the teleconnection patterns, the nature of the teleconnection pattern is still something of an enigma. In troposphere, the teleconnection pattern is usually horizontal wave–like and equivalent barotropic—for example, the Pacific–North America (PNA) pattern (Wallace and Gutzler 1981). With these features, the study of teleconnection pattern is related to the study of quasi-stationary Rossby wave, especially the quasi-stationary external Rossby wave (Held et al. 1985). Numerous studies have shown that the stationary Rossby waves could be considered as responses to the external forcing, such as the effects of orography (Charney and Eliassen 1949; Bolin 1950; Grose and Hoskins 1979; Hoskins and Karoly 1981; Held 1983) and thermal forcing (Smagorinsky 1953; Gill 1980; Hoskins and Karoly 1981; Held 1983; Ting and Held 1990; Ting and Yu 1998). However, the causes of low-frequency variability are diverse (Wallace and Blackmon 1983), such as forcing by high-frequency transients eddies (Held et al. 1989; Branstator 1992; Hoerling and Ting 1994; Branstator 1995), barotropic instability (Lorenz 1972; Simmons et al. 1983; Swanson 2000, 2001), wave–wave interactions (Longuet-Higgins et al. 1967; Gill 1974), and transitions between different climatic regimes (Charney and DeVore 1979; Charney and Straus 1980). These processes are associated with the internal variability of the atmosphere. Thus, to better understand and predict the teleconnection patterns, the effects of internal dynamics need to be studied.

During the last decade, a new teleconnection pattern along the Asian jet during boreal summer has been identified (Lu et al. 2002; Enomoto et al. 2003), which is named “the Silk Road pattern.” It is the leading mode of meridional wind variability in the upper troposphere over the Eurasian continent in boreal summer and has important influences on extratropical climate (Ding and Wang 2005; Ding et al. 2011; Chen and Huang 2012). The skillful seasonal prediction of the Silk Road pattern will be of great benefit to the economics, agriculture, and social life over the regions on its path. The previous studies have shown that the Silk Road pattern could be considered as stationary Rossby wave propagating along the Asian jet (Sato and Takahashi 2006; Kosaka et al. 2009), and this stationary Rossby wave is thermally driven by the Indian monsoon heating (Enomoto et al. 2003; Ding et al. 2011) or the thermal heating over the northwestern Indian Ocean according to Chen and Huang (2012). However, other studies have revealed the importance of internal dynamics to the Silk Road pattern. Sato and Takahashi (2006) found that barotropic kinetic energy conversion contributes to the self-strengthening of the Silk Road pattern. The research of Kosaka et al. (2009) indicates that baroclinic available potential energy conversion is more efficient for its self-maintenance. Nevertheless, there is no explicit mechanism to explain the self-maintenance of the Silk Road pattern. Therefore, in this study, the authors are attempting to provide such a mechanism.

The coexistence of the Silk Road pattern and the Asian jet suggests that the internal dynamics of the Silk Road pattern may be related to some kind of instability associated with the jet (e.g., baroclinic instability), as suggested by the energetics analysis of Kosaka et al. (2009). In the theory of the Charney instability problem, the stationary wave is marginally stable (Pedlosky 1987). A small perturbation of shear or wavenumber for this stationary wave will lead to instability and allows one to study the finite-amplitude dynamics of weakly unstable baroclinic disturbances by using a perturbation method. The finite-amplitude theories show that there exists an asymptotic steady state with a large wave amplitude in the presence of weak thermal damping and Ekman dissipation (Pedlosky 1979; Wang and Barcilon 1986). This may have implications for teleconnection patterns. However, the finite-amplitude theory for a continuous model is based on the Charney profile, which is different from the basic zonal wind profiles of the jet stream. Thus, the direct extension of finite-amplitude theory to the Silk Road pattern by using a continuous model with the Charney profile may not be appropriate. On the other hand, some studies have shown that the stationary wave with certain vertical structures could be destabilized by certain dissipation mechanisms. Held et al. (1986) have shown that the external Rossby wave could be destabilized by thermal damping and potential vorticity damping, which may have implications for theories of low-frequency variability in the extratropical troposphere. However, there is no observational implication in their study, and their theory is also based on the Charney wind profile. As an extension to the work of Held et al. (1986), with theoretical derivation and observational analysis, the authors will show here that the Silk Road pattern could self-maintain through baroclinic instability induced by thermal damping. The authors will also provide a mechanism to explain this kind of dissipation-induced baroclinic instability, which could be considered as a supplement of the mechanism proved by Held et al. (1986).

This paper is organized as follows. Section 2 will present some general features of the Silk Road pattern. By using observational analysis, the authors will examine the connection between the Silk Road pattern and stationary external Rossby waves. In section 3, the quasigeostrophic three-layer model is introduced, and the stationary external mode in this three-layer model is studied. In section 4, the baroclinic instability of the stationary external mode induced by thermal damping will be derived theoretically. The mechanism of this baroclinic instability will be discussed. As an observational implication, the potential energy conversion associated with the Silk Road pattern is also studied. Section 5 gives conclusions and a discussion.

## 2. Observation

In this section, we will present some general features of the Silk Road pattern and the observational implication of its connection to the stationary external Rossby waves. The dataset used in this section is the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) atmospheric reanalysis monthly data (Uppala et al. 2005). The data are chosen from the period of 1958–2002. The climatological mean is defined as a 45-yr average, and the anomalies are the deviations from this climatological mean. In this study, July is chosen as a typical month to represent summer, and January is chosen to represent winter.

### a. The basic zonal flow

As mentioned in introduction, previous studies have indicated that the Silk Road pattern could be considered as a quasi-stationary Rossby wave propagating along the Asian jet. Thus, before presenting the features of the Silk Road pattern, it is necessary to discuss the basic flow which the teleconnection pattern is embedded in, because the properties of linear waves are largely determined by the basic states. The stationary waves are often studied in the context of wintertime zonal flow. However, the vertical profiles of basic zonal flows are different in summer and winter. The climatological means of the zonally averaged zonal wind of January and July in Northern Hemisphere are depicted in Figs. 1a and 1b. There are westerly jets in the midlatitude troposphere both during winter and summer. One significant difference between Figs. 1a and 1b is that the tropospheric westerly jet is much weaker and farther north in summer than in winter. Another significant difference is that the basic zonal wind in the extratropical stratosphere is westerly in winter, while it is easterly in summer. In the theory of linear waves, to consider the baroclinic effect, the latitudinal variations of the basic zonal flow are sometimes ignored for simplicity and only the vertical variation is considered. Figures 1c and 1d depict the vertical profiles of basic zonal wind at certain latitudes corresponding to Figs. 1a and 1b. Figure 1c indicates that the vertical profile of the basic zonal wind at 60°N in winter is linearly increasing. This is the well-known Charney profile (Charney 1947), which has been widely used in theoretical works. The vertical profile at 30°N in winter is jetlike with a maximum in the upper troposphere. It is interesting to note that there is no Charney profile in summer. In summer, the vertical profiles of basic zonal flow are jetlike in both mid- and high latitudes. The difference is that the jet is much stronger in midlatitudes than in high latitudes. The basic zonal wind profiles increase linearly from the lower boundary to the upper troposphere and then decrease approximately linearly in the stratosphere. Consequently, the jetlike profile would be more suitable than the Charney profile for the theoretical study of the stationary Rossby wave in the summertime jet stream.

### b. The Silk Road pattern

To identify the Silk Road pattern, empirical orthogonal function (EOF) analysis is performed on 200-hPa meridional wind velocity anomalies in July over the region of 30°–60°N, 30°–130°E. The first EOF mode is the so-called Silk Road pattern. It explains 30.6% of the total variance. Since the midlatitude atmospheric motions satisfy the geostrophic balance, to better exhibit the horizontal structure of the EOF pattern, the regressions of 200-hPa geopotential height anomalies to the leading principal components (PCs) are calculated and the results are shown in Fig. 2a. The pattern shown in Fig. 2a is wavelike and zonally organized. The associated T–N flux (Takaya and Nakamura 2001) is also shown in Fig. 2a by vectors. The T–N flux is calculated by using regressed data, and it indicates the propagation of the Silk Road pattern. The Silk Road pattern originates at the entrance region of the Asian jet over the Mediterranean and Caspian Seas, propagates zonally along the core region of the Asian jet stream, with three significant anomalous centers over central Europe, western Asia, and central Asia, and then decays at the exit region of the Asian jet. The 200-hPa climatological rotational zonal wind velocity is shown in Fig. 2b. It is indicated by Figs. 2a and 2b that the Silk Road pattern is trapped longitudinally by the Asian jet, which could be considered as a strong waveguide in boreal summer (Hoskins and Ambrizzi 1993). This trapping by the Asian jet is the most distinctive feature of the Silk Road pattern.

Regressions of geopotential height anomalies (a) at 200 hPa and (c) along 45°N onto PC1 of EOF analysis based on the 200-hPa meridional wind velocity anomalies over the region of 30°–60°N, 30°–130°E. The vectors are the associated T–N flux obtained by regressed data. (b) The climatological mean of the 200-hPa rotational zonal wind velocity. The contour intervals are 5 m in (a), 10 m s^{−1} in (b), and 5 m in (c).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Regressions of geopotential height anomalies (a) at 200 hPa and (c) along 45°N onto PC1 of EOF analysis based on the 200-hPa meridional wind velocity anomalies over the region of 30°–60°N, 30°–130°E. The vectors are the associated T–N flux obtained by regressed data. (b) The climatological mean of the 200-hPa rotational zonal wind velocity. The contour intervals are 5 m in (a), 10 m s^{−1} in (b), and 5 m in (c).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Regressions of geopotential height anomalies (a) at 200 hPa and (c) along 45°N onto PC1 of EOF analysis based on the 200-hPa meridional wind velocity anomalies over the region of 30°–60°N, 30°–130°E. The vectors are the associated T–N flux obtained by regressed data. (b) The climatological mean of the 200-hPa rotational zonal wind velocity. The contour intervals are 5 m in (a), 10 m s^{−1} in (b), and 5 m in (c).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

The vertical structure of the Silk Road pattern along 45°N is shown in Fig. 2c. It is indicated that the vertical structure of the Silk Road pattern is equivalent barotropic. The wave amplitude increases with height in the troposphere, with maximum amplitude in the upper troposphere, and then decays with height in the stratosphere. The vertical structure of the Silk Road pattern is similar to the vertical structure of the stationary external Rossby wave given by Held et al. (1985), based on calculations using the wintertime data. However, the vertical profile of the basic zonal wind in summer is different than that in winter (Fig. 1), which may affect the vertical structure and wavenumber of the external mode. Therefore, in order to find a connection between the Silk Road pattern and an external Rossby wave, it is necessary to examine the stationary external mode by using summertime data.

### c. The stationary external mode in summer

*a*is the radius of Earth,

*N*is the buoyancy frequency, and

Vertical structures of the stationary external mode and the Silk Road pattern. The solid line represents the vertical structure of stationary external mode at 45°N calculated by using summertime observational data. The magnitude is multiplied by 20. Other lines represent the vertical structure of the Silk Road pattern at 37.5°, 65°, and 100°E along 45°N.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Vertical structures of the stationary external mode and the Silk Road pattern. The solid line represents the vertical structure of stationary external mode at 45°N calculated by using summertime observational data. The magnitude is multiplied by 20. Other lines represent the vertical structure of the Silk Road pattern at 37.5°, 65°, and 100°E along 45°N.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Vertical structures of the stationary external mode and the Silk Road pattern. The solid line represents the vertical structure of stationary external mode at 45°N calculated by using summertime observational data. The magnitude is multiplied by 20. Other lines represent the vertical structure of the Silk Road pattern at 37.5°, 65°, and 100°E along 45°N.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

## 3. Three-layer model

Since the basic zonal wind profile in midlatitudes during summer is jetlike, the problem will be complicated if the continuous model is used to study the stationary external mode. The reason is that the eigenvalue problem is difficult to solve analytically for a complex basic-flow profile (e.g., jetlike profile). However, the problem will be tractable if the layer model is used, because of its simplicity despite the reduction in vertical resolution. To represent the jetlike basic zonal flow, the model with three or more layers is necessary. Hence, for maximum simplification, the quasigeostrophic three-layer model will be introduced in this section, and the stationary external Rossby waves in this three-layer model will be studied.

### a. Quasigeostrophic three-layer model

Figure 4 illustrates the venerable and conventional quasigeostrophic three-layer model. The model contains three layers. Each layer is homogeneous and incompressible, with depth *D* in the absence motion of each layer. The fluids are immiscible and stably stratified with constant density *x* direction, bounded laterally by rigid vertical sidewalls at *y* = 0, *L*, and above and below by rigid boundaries at *z* = 0 and 3*D*. The *β*-plane approximation is adopted in this study, which is important for large-scale atmospheric motions in midlatitudes. The lower layer in this model represents the lower troposphere, the middle layer represents the upper troposphere, and the upper layer represents the lower stratosphere.

Illustration of the quasigeostrophic three-layer model.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Illustration of the quasigeostrophic three-layer model.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Illustration of the quasigeostrophic three-layer model.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

*U*,

*L*, 3

*D*, and 3

*UD*/

*L*are characteristic scales for the horizontal velocity, the horizontal length and vertical lengths of the motion, and the vertical velocity, respectively. Further, we assume that the Rossby number (

*f*

_{0}is Coriolis parameter) is small and expand the variables in powers of

*y*= 0 and

*y*=

*L*, the meridional velocity must vanish, which implies the vanishing of

### b. Stationary external waves in three-layer model

*U*are the basic zonal winds in each layer, and are assumed to be constant. Using (3.11), the linear equations of (3.7) could be written as

_{i}*l*is an integer to satisfy (3.9). Substituting (3.14) into (3.12) and negating the time derivatives, we obtain the matrix eigenvalue problem

*B*,

*S*

_{1},

*S*

_{2},

*U*

_{2}) is multiplied by a constant. Therefore, without loss of generality, we set

*a*and

*b*:

*B*and

*U*

_{2}for different values of

*S*

_{1}while fixing

*S*

_{1}increases, the parameter domain with three propagating modes reduces, and there are at most two propagating modes for the case

*S*

_{1}.

Number of horizontally wavelike stationary modes in the three-layer model as a function of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Number of horizontally wavelike stationary modes in the three-layer model as a function of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Number of horizontally wavelike stationary modes in the three-layer model as a function of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Figure 6 shows the wavenumber *B* and *U*_{2} for different values of *S*_{1} while fixing *B*, the wavenumber decreases as *U*_{2} increases. For a given *U*_{2}, the wavenumber increases as *B* increases. This result may be explained as follows. According to Held et al. (1985), the stationary wavenumber of external mode could be expressed as *U*_{2} approximates to

Wavenumber of stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 20.0, 30.0, and 40.0 are plotted.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Wavenumber of stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 20.0, 30.0, and 40.0 are plotted.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Wavenumber of stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0, 20.0, 30.0, and 40.0 are plotted.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

*B*and

*U*

_{2}for different values of

*S*

_{1}while fixing

*S*

_{1}increases, the region with vertical structure

*S*

_{1}increases to unity (the three-layer Phillips' model), there is no region where the vertical structure satisfies

Division of two types of external modes (mode I: *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Division of two types of external modes (mode I: *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Division of two types of external modes (mode I: *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

In conclusion, the quasigeostrophic three-layer model introduced in this section well captures the essential dynamical features of the stationary external modes corresponding to the observations. The equivalent barotropic structure of the external mode (

## 4. Baroclinic instability induced by thermal damping

### a. Derivation of destabilization

*A*has the form

As discussed by Shepherd (1990) and restated by Swanson (2001), all instabilities must have vanishing pseudoenergy if the pseudoenergy is conserved because the vanishing pseudoenergy will satisfy the dual requirements of the conservation of pseudoenergy and exponential growth in all measures of disturbance amplitude (eddy energy and enstrophy). In this sense, the quasi-stationary waves at lowest order have the potential to be unstable if there are certain destabilization mechanisms.

*B*,

*S*

_{1},

*S*

_{2},

*U*

_{2}) is multiplied by a constant—the same as that (3.15) infers. Therefore, without loss of generality, we set

*B*and

*U*

_{2}, and the results are shown in Fig. 8a with positive value hatched. It is indicated by Fig. 8a, for

*U*

_{2}is less than 1.2. Figure 8a also exhibits the calculation results for

Region of stationary external mode destabilized by thermal damping. The unstable regions are hatched as functions of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Region of stationary external mode destabilized by thermal damping. The unstable regions are hatched as functions of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Region of stationary external mode destabilized by thermal damping. The unstable regions are hatched as functions of *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

The inequality (4.21) is calculated for different values of *S*_{1} while fixing *S*_{2} = 1. The results are shown in Figs. 8b–e. A close comparison of Figs. 7 and 8 infers that vertical structure having a warm ridge and a cold trough from the lower to the middle layer (

The corresponding growth rates scaled by thermal damping rate are shown in Fig. 9. As indicated by Fig. 9, given a *U*_{2}, for *B* increases. Therefore, the beta effect is to stabilize the flow. For *U*_{2}, the above conclusion still holds. For *U*_{2}, the above conclusion holds for *S*_{1}, there are maximum growth rates at the range of *B*, the growth rate increases as *U*_{2} decreases. For given vertical wind shears, decreasing of *U*_{2} means increasing baroclinicity of the basic flow. Thus, the eddy grows more rapidly in more baroclinic basic flow. It can be seen from Fig. 9 that the growth rates increase rapidly as *U*_{2} is approaching the critical line of instability for *B* < 1.

Temporal growth rate for stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0, 4.0, 6.0, 8.0, and 10.0 are plotted. The stable regions are hatched.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Temporal growth rate for stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0, 4.0, 6.0, 8.0, and 10.0 are plotted. The stable regions are hatched.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Temporal growth rate for stationary external mode in three-layer model; *B* and *U*_{2}, for *S*_{2} = 1 and (a) *S*_{1} = *−*1, (b) *S*_{1} = −0.5, (c) *S*_{1} = 0, (d) *S*_{1} = 0.5, and (e) *S*_{1} = 1. The contours of 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0, 4.0, 6.0, 8.0, and 10.0 are plotted. The stable regions are hatched.

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

### b. Mechanism of destabilization

*i*th surface from its undisturbed value (Pedlosky 1987),

The physical meaning of this destabilization effect could be understood as follows. Given the equivalent barotropic structure of the first order (e.g.,

### c. Observational implications

Although the theory derived above may explain the self-maintenance of the Silk Road pattern, the extension of this theory to the observations must be cautious, because of the uncertain connection of this simplistic three-layer model to the real atmosphere. On the other hand, the clear physical meaning and rich dynamical features that this three-layer model displays make this extension possible. However, it is necessary to discuss how the channel model could be applied to explain the self-maintenance of Silk Road pattern before any extension to observation. As can be seen in Figs. 2a and 2b, the Silk Road pattern appears to be trapped by Asian jet waveguide. The Asian jet waveguide confines the Silk Road pattern in a channel between 30° and 50°N, and the zonal scale of this channel is much larger than the wavelength. Thus, it is reasonable to use a channel model to model this teleconnection pattern. In this channel, although the zonal wind exhibits zonal variations, it varies slowly with longitude compared with the scale of the wavelength. Therefore, it may be a good approximation by using zonally averaged zonal wind between 10° and 160°E as basic flow of this channel model if we assume the zonal extension the channel is from 10° to 160°E. Therefore, the three-layer channel model with zonal symmetric basic zonal flow is suited for the study of the Silk Road pattern.

To extend the theory to the real atmosphere, the heat flux associated with the Silk Road pattern needs to be examined, since the baroclinic energy conversion between the eddy and the basic flow is a consequence of the instability according to the theory. However, it must be noted that the potential energy conversion is just the necessary condition for instability, not the sufficient condition, as demonstrated by Farrell (1985) that transient baroclinic development in the absence of baroclinic instability can also extract potential energy from the mean flow. Therefore, the potential energy conversion could only suggest the applicability of above theory in explaining the self-maintenance of the Silk Road pattern.

*S*is the static stability,

*f*is the Coriolis parameter, and

*T*is the temperature. The vertical coordinate in log-pressure coordinates

*z*is defined as

*p*

_{s}is taken to be 1000 hPa. The density

*z*= 0 and is taken as 1.225 kg m

^{−3}.

It can be inferred from (4.30) that if the meridional heat flux

(a) Latitude–pressure cross section of meridional heat flux (W m^{−2}) associated with the Silk Road pattern averaged between 10° and 160°E (shading), and climatological mean of rotational zonal flow averaged between 10° and 160°E. The contour interval is 4 m s^{−1}. (b) Vertical integration of baroclinic energy conversion (W m^{−2}).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

(a) Latitude–pressure cross section of meridional heat flux (W m^{−2}) associated with the Silk Road pattern averaged between 10° and 160°E (shading), and climatological mean of rotational zonal flow averaged between 10° and 160°E. The contour interval is 4 m s^{−1}. (b) Vertical integration of baroclinic energy conversion (W m^{−2}).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

(a) Latitude–pressure cross section of meridional heat flux (W m^{−2}) associated with the Silk Road pattern averaged between 10° and 160°E (shading), and climatological mean of rotational zonal flow averaged between 10° and 160°E. The contour interval is 4 m s^{−1}. (b) Vertical integration of baroclinic energy conversion (W m^{−2}).

Citation: Journal of the Atmospheric Sciences 70, 9; 10.1175/JAS-D-12-0326.1

Even though the observed structure of meridional heat flux implies the potential baroclinic instability, one might still question whether thermal damping truly destabilized the Silk Road pattern. However, according to Hendon and Hartmann (1982), the latent heat could distribute the sensible heat flux (which has the same form of Newtonian cooling) higher into the atmosphere. They find that the sensible heat flux with deep distribution could amplify the stationary waves. Thus, the latent heat associated with extratropical waves could increase the potential for amplifying external modes by distributing the thermal damping through a great depth (Held et al. 1986). In the study of Chen and Huang (2012), they found that the Silk Road pattern could induce precipitation anomalies over Eurasian continent and thus was related to deep distribution of thermal damping. Therefore, it implies that the instability considered in this paper could be considered as a plausible mechanism for the self-maintenance of Silk Road pattern.

## 5. Discussion and conclusions

In this paper, the internal dynamics of the Silk Road pattern has been studied. Since observation indicates that the Silk Road pattern could be considered as stationary external Rossby waves, the quasigeostrophic three-layer model has been used to study the dynamics of the external Rossby waves. The three-layer model well captures the essential dynamical features of the stationary external Rossby waves in accordance with the observations. Theoretical analysis indicates that the quasi-stationary external modes could be destabilized by weak thermal damping. For destabilization to occur, the vertical structures of the external modes must be equivalent barotropic. The effect of thermal damping could be considered as modifying the eddy streamfunction in such way that the eddy streamfunction has vertical phase tilt, so the eddy could feed on the basic zonal flow by extracting the potential energy. The potential energy conversion of this instability is similar to the classic baroclinic instability in the sense that the eddy obtains potential energy from the basic zonal flow, except that the instability in this study is induced by the weak thermal damping. The observational analysis of the baroclinic energy conversion indicates that the Silk Road pattern could extract potential energy from the basic flow and the structure of meridional heat flux agrees with the theory to a certain degree. Therefore, it suggests that the theory in study could serve to explain the self-maintenance of the Silk Road pattern. However, there may be other mechanisms to explain the self-maintenance of the Silk Road pattern, since the baroclinic energy conversion does not assure the baroclinic instability (Farrell 1985). Nevertheless, as discussed in section 4c, the baroclinic instability induced by thermal damping could be considered as a plausible mechanism for the self-maintenance of Silk Road pattern.

In many studies, observational analysis and numerical investigations reveal that the Silk Road pattern could be considered as a extratropical response to the external thermal heating, such as heating of the Indian summer monsoon (Enomoto et al. 2003; Ding and Wang 2005; Ding et al. 2011) or heating over the northern Indian Ocean (Chen and Huang 2012). In this study, using observational analysis and theoretical derivations, the authors have shown that the Silk Road pattern could be self-maintained by baroclinic instability induced by weak thermal dissipation. As a consequence, both the external forcing and internal dynamics produce an effect on the Silk Road pattern. Once the Silk Road pattern is forced by the external thermal heating, it could be self-strengthened through the internal dynamics while traveling away from the heating sources. The combined effect of the external forcing and internal dynamics makes the Silk Road pattern very robust in boreal summer, which is manifested by the fact that the Silk Road pattern is the leading mode of meridional wind variability at the upper troposphere over the Eurasian continent in summer. The result of this study may have significant implications, because it provides an interpretation of the extratropical teleconnection patterns from a new perspective. As shown in Figs. 1c and 1d, the vertical profiles of the basic zonal wind in the extratropical troposphere are usually a Charney profile or a jetlike profile. In section 5, we have shown that external modes associated with these kinds of zonal wind shear could be destabilized by thermal dissipation. Therefore, in the general sense, besides the Silk Road pattern, the extratropical teleconnection patterns with equivalent barotropic structure could also be self-maintained through this mechanism. However, it should be noted that the successful extensions to other cases depend on the vertical profile of the thermal damping rate (Held et al. 1986; Robinson 1987) and the stabilization effect of Ekman pumping (it has been verified by the authors that the Ekman pumping could reduce the destabilization).

In this study, only the linear dynamics are considered. In linear theory, the eddy amplitude grows exponentially. Thus, the eddy amplitude will eventually become large enough so that the linear theory breaks down. Consequently, there must be a bound to the eddy amplitude or the eddy energy when the eddy amplitude is large, and the nonlinear effects must be considered. The upper bound of eddy energy may be estimated by using the method proposed by Mu et al. (1994). However, the nonlinear saturation problem in this case is rather complicated and awkward. On the other hand, it is also interesting whether there may exist any asymptotic state when the weakly nonlinear effects are considered, as in the continuous problems (Pedlosky 1979; Wang and Barcilon 1986). Therefore, further study on nonlinear effects is needed.

## Acknowledgments

This study was supported by the National Nature Science Foundation of China (Grant 41230527) and the National Basic Research Program of China (Grant 2010CB950403).

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