Abstract

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 external Rossby waves. The three-layer model well captures the essential dynamical features of stationary external Rossby waves in accordance with the observations. Theoretical analysis indicates that the quasi-stationary external modes could be destabilized by the weak thermal damping. For destabilization to occur, the vertical structures of the external modes must have a warm ridge and a cold trough from the lower to middle layers. The effect of thermal damping could be considered as modifying the eddy streamfunction in such way that the eddy streamfunction has a vertical phase tilt, so the eddy could feed on the basic zonal flow by extracting the potential energy. The implications for this baroclinic instability on the self-maintenance of the Silk Road pattern are discussed. The observations imply that this dissipative destabilization mechanism could explain the self-maintenance of the Silk Road pattern.

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.

Fig. 1.

Climatological mean of the zonally averaged zonal wind velocity in (a) January and (b) July. The contour intervals are 10 m s−1 in (a) and 4 m s−1 in (b). (c) The corresponding vertical profiles of zonal wind at 30° and 60°N in (a). (d) The corresponding vertical profiles of zonal wind at 45° and 70°N in (b).

Fig. 1.

Climatological mean of the zonally averaged zonal wind velocity in (a) January and (b) July. The contour intervals are 10 m s−1 in (a) and 4 m s−1 in (b). (c) The corresponding vertical profiles of zonal wind at 30° and 60°N in (a). (d) The corresponding vertical profiles of zonal wind at 45° and 70°N in (b).

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.

Fig. 2.

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).

Fig. 2.

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).

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

Following Held et al. (1985), considering the eigenvalue problem at latitude ,

 
formula
 
formula

where a is the radius of Earth, indicates discretized vertical mode, is the meridional wind velocity, is the Coriolis parameter, N is the buoyancy frequency, and is the zonally averaged quasigeostrophic vorticity. The meridional gradient of and zonally averaged relative vorticity can be written as

 
formula

Assuming that the basic zonal flow varies sufficiently slowly with latitude, so the vertical modes and horizontal modes can be separated. The total wavenumber of the horizontal structure equation is then . For horizontally wavelike mode, is negative. The nondimensional total horizontal wavenumber could be defined as

 
formula

The calculation procedure used by Held et al. (1985) is adopted. The centered finite-differencing scheme is used for (2.1) and (2.2). We only calculate the external mode at 45°N. The 14 pressure levels (1000, …, 70 hPa) are chosen where at all levels. The calculation shows that there is one and only one mode that is horizontally wavelike (), which implies that the horizontally wavelike disturbances perturbed about the midlatitude jet stream during summer must be barotropic equivalent. The nondimensional wavenumber of the external mode is 6.668, which agrees with the wavenumber of the Silk Road pattern shown in Fig. 2a, and is also consistent with the stationary barotropic Rossby wavenumber in the summertime Asian jet (Kosaka et al. 2009). The vertical structure of the external mode is shown in Fig. 3. The vertical profiles of the Silk Road pattern at 37.5°, 65°, and 100°E along 45°N are also shown in Fig. 3. It is indicated that the vertical structure of the Silk Road pattern is consistent with the stationary external mode. Since both the wavenumber and vertical structure of the Silk Road pattern are consistent with the stationary external mode, it is inferred that the Silk Road pattern could be considered as a stationary external Rossby wave. Hence, to study the internal dynamics of the Silk Road pattern, we need to study the dynamics of the stationary external mode.

Fig. 3.

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.

Fig. 3.

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.

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 in each layer, which satisfies the relation . We assume that density differences and are both equal to and are much less than the characteristic density . The fluids are confined in a channel infinite in extent in the x direction, bounded laterally by rigid vertical sidewalls at y = 0, L, and above and below by rigid boundaries at z = 0 and 3D. 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.

Fig. 4.

Illustration of the quasigeostrophic three-layer model.

Fig. 4.

Illustration of the quasigeostrophic three-layer model.

The derivation of the quasigeostrophic equations in this three-layer model was described in previous studies (Davey 1977; Moroz 1981; Pedlosky 1987) and we merely summarize the essential details. Suppose that U, L, 3D, and 3UD/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 (, where f0 is Coriolis parameter) is small and expand the variables in powers of ; for example,

 
formula

At the lowest order , the motions satisfy the geostrophic balance

 
formula

At the order , we obtain the nondimensional equations of motion

 
formula

where is the two-dimensional Jacobian, are the streamfunctions in each layer, and is the quasigeostrophic potential vorticity

 
formula

where is the internal rotational Froude number and is the nondimensional measure of the beta effect ( is the corresponding dimensional variable). The matrix is given by

 
formula

We further decompose the fluids into basic states and disturbances:

 
formula

Assuming the disturbance is an order smaller than the basic flow and substituting (3.6) into (3.3), we obtain the equations of disturbances

 
formula

where is the perturbation quasigeostrophic potential vorticity. The meridional gradient of the basic quasigeostrophic potential vorticity is

 
formula

On the lateral boundary of y = 0 and y = L, the meridional velocity must vanish, which implies the vanishing of ,

 
formula

and the vanishing of ,

 
formula

b. Stationary external waves in three-layer model

To study the stationary external Rossby wave, the basic flow is set to be

 
formula

where Ui 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

 
formula

where

 
formula

Writing stationary normal-mode solutions in vector form,

 
formula

where c.c. denotes the complex conjugate and 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

 
formula

where

 
formula

As can be inferred from (3.15), the solutions of (3.15) are invariant if (B, S1, S2, U2) is multiplied by a constant. Therefore, without loss of generality, we set to represent the jetlike basic zonal wind in summer.

For , if we define

 
formula

then the eigenvalues of (3.15) could be expressed in terms of a and b:

 
formula

It is easy to prove that is the largest eigenvalue and thus is the wavenumber of stationary external Rossby wave. It can be proved that is positive if and only if , which infers that the stationary external mode will be horizontally wavelike if and only if the basic zonal winds are positive in all layers. If and , then is also positive and there are two horizontally wavelike modes. If and , then will be positive and there are three horizontally wavelike modes. Figure 5 shows the number of horizontally wavelike modes as a function of B and U2 for different values of S1 while fixing by solving the eigenvalue problem of (3.15) numerically. As indicated by Fig. 5a, the calculations support the above derivations. As depicted by Fig. 5, in most of the parameter domains, there is one and only one horizontally wavelike mode and the external mode is always horizontally wavelike. This means, in general, there will be no horizontally wavelike modes except the external modes, which is consistent with the observations. It is also indicated by Fig. 5 that as S1 increases, the parameter domain with three propagating modes reduces, and there are at most two propagating modes for the case . The parameter domain of one propagating mode remains invariant for different S1.

Fig. 5.

Number of horizontally wavelike stationary modes in the three-layer model as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 1.

Fig. 5.

Number of horizontally wavelike stationary modes in the three-layer model as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 1.

Figure 6 shows the wavenumber of the external mode as function of B and U2 for different values of S1 while fixing . As indicated by Fig. 6, for a given B, the wavenumber decreases as U2 increases. For a given U2, 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 , where the angle bracket denotes vertical integration and is the meridional wind velocity of external mode. If we define , then the wavenumber can be written as , which is identical to the stationary barotropic Rossby wavenumber. Generally, U2 approximates to , and thus we have . Therefore, the results of Fig. 6 are explained.

Fig. 6.

Wavenumber of stationary external mode in three-layer model; is plotted as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 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.

Fig. 6.

Wavenumber of stationary external mode in three-layer model; is plotted as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 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.

Despite its reduction in vertical resolution, one still expects that the three-layer model could produce reasonable vertical structure of the external mode. For the jetlike profile , the vertical structure function of the external Rossby waves could be solved directly from (3.15):

 
formula

Therefore, corresponding to the equivalent barotropic structure of stationary external modes in observation (), must be larger than unity. This condition will be satisfied if and only if

 
formula

For different vertical shears of the basic zonal flow, the vertical structure of the external stationary Rossby wave will be modified. Figure 7 shows the division of two types of equivalent barotropic external modes ( and ) as function of B and U2 for different values of S1 while fixing . These two types of vertical structures have a warm ridge and a cold trough from the lower to the middle layers. As will be demonstrated in section 4, the external modes with these two vertical structures could be destabilized by thermal damping. As S1 increases, the region with vertical structure reduces. As S1 increases to unity (the three-layer Phillips' model), there is no region where the vertical structure satisfies . The vertical structure with a warm ridge and a cold trough from the lower to the middle layers has only the type . These modifications of vertical structure are due to the reduction in the model's ability to resolve the vertical structure of the motion and the vertical profiles of the basic zonal wind. Comparing to the layer model, the amplitude of the external mode will ultimately decay with height in the continuous model, even if the basic zonal wind is linearly increasing in the vertical direction (Held et al. 1985).

Fig. 7.

Division of two types of external modes (mode I: and mode II: , corresponding to modes with a warm ridge and a cold trough from the lower to the middle layer) in a three-layer model as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 1.

Fig. 7.

Division of two types of external modes (mode I: and mode II: , corresponding to modes with a warm ridge and a cold trough from the lower to the middle layer) in a three-layer model as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 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 () is produced in this three-layer model. In the next section, we will demonstrate this external mode could be destabilized by thermal dissipation under certain conditions.

4. Baroclinic instability induced by thermal damping

a. Derivation of destabilization

Considering the quasi-stationary aspect of the Silk Road pattern, we introduce a “slow” time variable

 
formula

where is a nondimensional parameter and is assumed to be sufficiently small. Consider asymptotic expansion of the disturbances in powers of :

 
formula

If we substitute (4.2) into (3.7), then at the lowest order, the equations will just be (3.15). Thus, the stationary external modes could be represented as disturbances at the lowest order. In this section, we will demonstrate that the external mode at the lowest order could be destabilized by thermal dissipation.

First, we will consider the conservation law of pseudoenergy in the three-layer model, since the conservation of pseudoenergy could lead to the necessary condition for instability (McIntyre and Shepherd 1987; Shepherd 1990). If we consider Newtonian thermal cooling effect and linearize (3.7), we obtain

 
formula

where is the thermal damping rate and is assumed to be constant. Following the derivations of Vallis (2006), the pseudoenergy conservation law of this system could be written in the form

 
formula

where the pseudoenergy A has the form

 
formula

The overbar represents the zonal average, which has the form .

The dissipation of pseudoenergy by thermal damping has the form

 
formula

We suppose

 
formula

Substitute (4.2) into (4.4), and use the fact that

 
formula

Then, after a little algebra, the equation of (4.4) is

 
formula

where

 
formula

It is surprising to note that the pseudoenergy at the lowest order is conserved even in the presence of the thermal damping. In other words, the pseudoenergy of the quasi-stationary Rossby wave at the lowest order does not feel the dissipative effect of thermal damping.

Substitute (3.14) into (4.10), and we have

 
formula

Note that

 
formula

Substitute (4.12) into (4.11), and, after some algebra, we obtain

 
formula

It can be proved that is zero after integration. Thus, the pseudoenergy of quasi-stationary wave at the lowest order is vanishing. This is consistent with similar result of Plumb (1985) that stationary waves must have vanishing pseudoenergy. It is noted that the vanishing of pseudoenergy requires the conservation law of (4.9); otherwise, the pseudoenergy would become nonvanishing.

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.

Although the analysis of pseudoenergy reveals the potential instability of the quasi-stationary waves at the lowest order, it does not provide an explicit form of instability. So we will explicitly derive the baroclinic instability of stationary external mode induced by thermal damping. Following Held et al. (1986), multiplying (4.3) by , and integrating vertically and horizontally, we can obtain the following pseudomomentum conservation law:

 
formula

We assume that the solutions have the form

 
formula

Corresponding to (4.2), we expand the vertical structure and frequency of the disturbances in the powers of :

 
formula

where is zero for the stationary waves. Then, at the lowest order, the perturbation streamfunction has the form

 
formula

Note that the vertical structure function of the disturbances at the lowest order in (4.17) satisfies (3.15). Substituting (4.17) into (4.14), we can obtain the approximation

 
formula

The solutions of (4.18) will be invariant if (B, S1, S2, U2) is multiplied by a constant—the same as that (3.15) infers. Therefore, without loss of generality, we set to represent the summertime jetlike basic flow.

For the external modes in jetlike basic flow, using (3.19), (4.18) can be further simplified as

 
formula

Thus, for perturbations to grow, it is necessary that

 
formula

In general, the expression in the second square bracket will be positive if the inequality (3.20) is satisfied. Thus, to satisfy (4.20), the expression in the first square bracket must also be positive. It is obvious that this will be satisfied if . For , the calculations are processed. The expression on the left-hand side of (4.20) is calculated as functions of B and U2, and the results are shown in Fig. 8a with positive value hatched. It is indicated by Fig. 8a, for , the inequality (4.20) is satisfied in most of the parameter domain, except for a small region where U2 is less than 1.2. Figure 8a also exhibits the calculation results for , which is consistent with the above derivations.

Fig. 8.

Region of stationary external mode destabilized by thermal damping. The unstable regions are hatched as functions of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 1.

Fig. 8.

Region of stationary external mode destabilized by thermal damping. The unstable regions are hatched as functions of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 1.

For other basic zonal wind shears, the destabilization condition is

 
formula

The inequality (4.21) is calculated for different values of S1 while fixing S2 = 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 ( or ) is a necessary condition for the destabilization. This agrees with the similar results of Held et al. (1986). It is also indicated in Fig. 8 that the external modes could be destabilized for a wide choice of parameters in all cases, which implies that the destabilization process is apt to occur in the real atmosphere.

The corresponding growth rates scaled by thermal damping rate are shown in Fig. 9. As indicated by Fig. 9, given a U2, for , the growth rate decreases as B increases. Therefore, the beta effect is to stabilize the flow. For and large U2, the above conclusion still holds. For and small U2, the above conclusion holds for , while for other values of S1, there are maximum growth rates at the range of . On the other hand, for a given B, the growth rate increases as U2 decreases. For given vertical wind shears, decreasing of U2 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 U2 is approaching the critical line of instability for B < 1.

Fig. 9.

Temporal growth rate for stationary external mode in three-layer model; is plotted as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 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.

Fig. 9.

Temporal growth rate for stationary external mode in three-layer model; is plotted as a function of B and U2, for S2 = 1 and (a) S1 = 1, (b) S1 = −0.5, (c) S1 = 0, (d) S1 = 0.5, and (e) S1 = 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.

b. Mechanism of destabilization

To better understand the physics, the mechanism of this destabilization will be studied. Multiply (4.3) by and integrate over the domain, and we obtain the eddy energy conservation law:

 
formula

which can be further written as

 
formula

As indicated by (4.23), the thermal dissipation will ultimately damp the eddy energy. Thus for eddy amplitude to grow, it must extract potential energy from the basic zonal flow through the first term on the right-hand side of (4.23). This baroclinic energy conversion process is identical to the classic baroclinic instability.

Since the potential energy conversion is the only source for wave to grow, the eddy thermal equations in the three-layer model is then studied. Consider the thermal equations in the three-layer model:

 
formula

where is the nondimensional deviation of the ith surface from its undisturbed value (Pedlosky 1987), is the vertical velocity, and represents Newtonian cooling. Using the notation of , the first term on the right-hand side of (4.23) could be written as

 
formula

Multiplying by on both sides of (4.24) and integrating in zonal direction, the equation of zonally averaged eddy potential energy is

 
formula

Substituting (4.1) into (4.26), we obtain the equation of (4.26)

 
formula

and the equation

 
formula

where .

If the normal-mode solution (4.17) is substituted into , then the zonally averaged meridional heat flux at the lowest order could be expressed as

 
formula

It is implied by the last term on the right-hand side of (4.26) that thermal dissipation will damp the eddy potential energy, which is consistent with (4.23). The second term on the right-hand side of (4.26) represents the conversion between eddy potential energy and eddy kinetic energy. The first term on the right-hand side of (4.26) represents potential energy conversion between the basic zonal flow and the eddy, and is responsible for the instability. At order 1, these two energy conversion terms are balanced, as manifested by (4.27). That is, at the lowest order, the eddy obtains potential energy from basic flow and then transfers it to kinetic energy, and vice versa. It can be verified that the meridional heat flux at lowest order is vanishing [it could be verified that the right-hand side of (4.29) is vanishing], which could be explained since the external mode has no phase tilt and thus has no zonally averaged meridional heat flux. However, , must not be vanishing, otherwise the eddy could not extract potential energy and grow, as manifested by (4.23) and (4.28). Since the correction to the eddy is related to the thermal damping, as a result, the effect of thermal damping could be considered as causing basic flow to release of potential energy and feed the eddy.

The physical meaning of this destabilization effect could be understood as follows. Given the equivalent barotropic structure of the first order (e.g., , corresponding to the observation), there are warm temperature anomalies at the lower boundary and cold temperature anomalies at the upper boundary associated with the perturbation ridge. According to Held et al. (1986) and Robinson (1987), the thermal damping could be thought of as shifting the warm temperature anomaly at the lower boundary westward and the cold temperature anomaly at the lower boundary eastward, relative to the geopotential ridge in the middle. This shifting of temperature anomalies at the lower and upper boundary will lead to a westward phase tilt of streamfunction from the lower to middle layer and eastward phase tile from the middle to upper layer. As a result, the thermal damping could be thought of as modifying the streamfunctions in such a way that the streamfunction has a vertical phase tilt, so the wave could extract potential energy from basic flow and convert to wave kinetic energy. This effect of thermal damping is manifested in the presence of second-order solutions. It should be noted that it is the superposition of the first- and second-order streamfunctions that has a vertical phase tilt. Moreover, as the second-order solution is an order smaller than the first-order solution, this vertical phase tilt is only a small departure from the barotropic equivalent structure of the first-order solution.

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.

The data used in the following analysis are the regressions of atmosphere variables to the PC associated with Silk Road pattern obtained in section 2. The basic state is chosen as the climatological rotational zonal wind averaged between 10° and 160°E. Before presenting the observational implication, it is noted that in three-layer model is proportional to in the continuous model. Corresponding to (4.25), the dimensional potential energy conversion between the basic flow and the eddy in continuous model in log-pressure coordinate (Holton 2004) can be written as

 
formula

where 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 , where ps is taken to be 1000 hPa. The density has the form , where is the density at z = 0 and is taken as 1.225 kg m−3.

It can be inferred from (4.30) that if the meridional heat flux is positive in the region of positive vertical zonal wind shear , and negative if , then the eddy will extract potential energy and feed on the basic flow. However, it should be noted that since the lower atmosphere is denser than the upper atmosphere, the vertical integration of (4.30) may be positive even if the meridional heat flux is positive in all regions. Nevertheless, according to the discussion in section 4b, it requires the meridional heat flux to change signs between the upper and lower atmosphere for disturbance, which has the form of . The zonally averaged meridional heat flux between 10° and 160°E associated with the Silk Road pattern is calculated and the results are shown in Fig. 10a. The basic zonal wind averaged between 10° and 160°E is also plotted in Fig. 10a. It is indicated that the observed meridional heat flux matches the theory to a certain degree, although in some regions there is positive heat flux where the basic flow shear is negative. As a result, it suggests that the baroclinic instability theory derived in this paper could be served to explain the self-maintenance of Silk Road pattern. Since (4.30) is written as a vertical integral form, to be more convincing, the vertical integration is calculated. Figure 10b shows the vertical integration of as a function of latitude. It indicates that the Silk Road pattern could extract potential energy from the basic flow and be self-maintained.

Fig. 10.

(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).

Fig. 10.

(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).

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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