## 1. Introduction

Blocking anticyclonic flows have long been recognized to have a profound impact on regional weather and climate, not only over the areas in which they occur, but also over upstream and downstream regions through their modification of synoptic-scale disturbances (Berggren et al. 1949). On the other hand, Green (1977) suggested that baroclinic synoptic-scale disturbances might act to maintain a planetary-scale blocking circulation, a notion that has drawn strong support from numerical experiments (Shutts 1983; Haines and Marshall 1987). Thus, considerable attention has been focused on the two-way relationship between blocking anticyclones and midlatitude baroclinic disturbances (Colucci 1985; Holopainen and Fortelius 1987; Nakamura and Wallace 1993). Figure 1 illustrates the two-way relationship between blocks and synoptic-scale disturbances over the North Pacific (Higgins and Schubert 1994). It is clear that an enhancement of synoptic-scale disturbances coincides with block onset and that a northward shift of the Pacific storm track lasts throughout the maintenance phase of blocking.

Colucci (1985) proposed that the process leading to block formation is a nonlinear interaction between ultralong waves and intense cyclones, but certainly not every intense cyclone results in the formation of a blocking anticyclone. Tsou and Smith (1990) noted that block formation preceded by rapidly developing cyclones is the result of the superposition of mobile and amplifying synoptic-scale cyclones and a stationary planetary-scale ridge. Cai and Van den Dool (1991) have documented that high-frequency vorticity fluxes associated with the storm tracks act to amplify low-frequency waves and retard their propagation. In an observational study of block events, Orlanski (1998) demonstrated that, during El Niño years, the blocking ridge is weaker and the storm track extends toward the west coast of the United States whereas, during La Niña years, the blocking ridge is stronger and storms are more commonly deflected poleward into northern Canada and Alaska.

Unfortunately, cause and effect are difficult to infer from diagnostic analyses. In particular, it is not clear whether the storm tracks are as they are because the stationary diffluent flow (ridge) modulates them or if the diffluent flow is a response to the eddy forcing along the baroclinic storm track. Although several theoretical models have been proposed to explain the role of high-frequency eddies in exciting a blocking circulation (Haines and Marshall 1987; Malanotte-Rizzoli and Malguzzi 1987; Malguzzi 1993; Luo 2000), the following questions remain unresolved: 1) What roles do the self-interaction among synoptic-scale waves and the planetary–synoptic-scale interaction play in the life cycle of a blocking? 2) Why do atmospheric blocking anticyclones associated with synoptic-scale eddies exhibit a strong meandering flow composed of several isolated anticyclonic and cyclonic vortices (Berggren et al. 1949, their Figs. 14–18 and 26; Butchart et al. 1989)? 3) Why does the synoptic-scale wave activity tend to enhance and split into two branches before the blocking pattern is fully established (Nakamura and Wallace 1993)? 4) Why do the storm track eddies tend to deflect (shift) northward during the life cycle of blocking anticyclones (Holopainen and Fortelius 1987; Higgins and Schubert 1994; Orlanski 1998)?

The principal objective of the present paper is to propose a new theoretical model to address the above issues based upon two assumptions that are motivated by observations (Shepherd 1987) that (i) there exists a zonal-scale separation between the planetary-scale and synoptic-scale waves and (ii) the range of synoptic-scale zonal wavenumbers is comparable to the planetary-scale zonal wavenumber. These assumptions make the synoptic-scale evolution equation linear, while retaining the nonlinearity of the planetary-scale evolution equation, and allow an asymptotic treatment. While the resulting theory may not be quantitatively accurate for the atmosphere, the merit of an asymptotic theory is that it provides a self-consistent mathematical description of a phenomenon in a certain limit, which captures the essential physics. The proposed model is presented in the spirit of the forced nonlinear Rossby wave critical layer (Warn and Warn 1978; Stewartson 1978), another barotropic asymptotic model, which is now accepted as the paradigm for Rossby wave breaking in the stratosphere (McIntyre and Palmer 1983; Juckes and McIntyre 1987). Numerical simulations complement asymptotic theory by examining more realistic parameter regimes.

The plan of the paper is as follows. In section 2, the block–eddy interaction equations are derived and from them we obtain a forced nonlinear Schrödinger (NLS) equation describing the evolution of the planetary-scale blocking initiated by synoptic-scale activity using a multiple-scale expansion. The analytical solution for the feedback of the block evolution on preexisting synoptic-scale eddies is derived in section 3. Combining the two results, in section 4 the main characteristics of the interaction between an incipient block and preexisting synoptic-scale eddies are presented, and a possible cause for the northward deflection of storm tracks associated with block onset is discussed. Discussion and conclusions are presented in section 5.

## 2. Governing equations of block development associated with synoptic-scale eddies

### a. Model description

_{T}is the streamfunction,

*F*= (

*L*/

*R*

_{d})

^{2}, and

*β*=

*β*

_{0}

*L*

^{2}/

*U*; here

*R*

_{d}is the Rossby deformation radius,

*β*

_{0}is the meridional gradient of the Coriolis parameter;

*L,*

*U,*and

*h*

*J*(

*a,*

*b*) = (∂

*a*/∂

*x*)(∂

*b*/∂

*y*) − (∂

*a*/∂

*y*)(∂

*b*/∂

*x*) is the Jacobian operator; ∇

^{2}is the horizontal Laplacian operator;

*h** is a planetary-scale topographic variable;

*f*

_{0}is the fixed Coriolis parameter;

*D*is the mean depth of the fluid; and

*ψ*

^{*}

_{S}

_{T}into three parts: Ψ

_{T}=

*ψ*

_{0}+

*ψ*+

*ψ*′, where

*ψ*

_{0}= −

*u*

_{0}

*y*is the basic-state field (where

*u*

_{0}is the uniform basic westerly wind and a constant), and

*ψ*and

*ψ*′ are the planetary-scale wave, characterized by zonal wavenumber

*k*corresponding to the frequency

*ω,*and synoptic-scale waves, characterized by zonal wavenumbers

*k̃*

_{n}(

*n*= 1, 2) corresponding to the frequencies

*ω̃*

_{n}respectively. We now assume (i) that

*k*≪

*k̃*

_{n}and (ii) that the range of

*k̃*

_{n}, Δ

*k̃,*satisfies Δ

*k̃*∼

*k*≪

*k̃*

_{n}. These assumptions are idealized, but are motivated by observations (Shepherd 1987). In this case, the self-interaction term

*J*(

*ψ,*∇

^{2}

*ψ*) of the planetary wave lies within the planetary-scale range, while the planetary–synoptic-scale interaction terms

*J*(

*ψ*′, ∇

^{2}

*ψ*+

*h*) and

*J*(

*ψ,*∇

^{2}

*ψ*′) are found to approximately have zonal wavenumbers

*k̃*

_{n}±

*k*and frequencies

*ω̃*

_{n}±

*ω.*For quasi-stationary planetary-scale (zonal wavenumbers 1–2) wave

*ψ*satisfying |

*ω*| ≪ |

*ω̃*

_{n}|, we can have

*ω̃*

_{n}±

*ω*≈

*ω̃*

_{n}. In this case,

*J*(

*ψ*′, ∇

^{2}

*ψ*+

*h*) and

*J*(

*ψ,*∇

^{2}

*ψ*′) are approximately in the synoptic-scale range. Furthermore, the synoptic-scale interaction term, denoted by

*J*(

*ψ*′, ∇

^{2}

*ψ*′)

_{P}, is a projection onto the planetary scale, while the projection of the synoptic-scale component, denoted by

*J*(

*ψ*′, ∇

^{2}

*ψ*′)

_{S}, onto the synoptic scales near

*k̃*

_{n}is negligible because it has zonal wavenumber

*k̃*

_{1}+

*k̃*

_{2}(

*k̃*

_{1}+

*k̃*

_{2}≫

*k̃*

_{n}). Thus,

*J*(

*ψ*′, ∇

^{2}

*ψ*′)

_{S}, which lies remarkably outside the (narrow) synoptic-scale range, can be ignored in the synoptic-scale equation. In this case, the following equations for the interaction between planetary- and synoptic-scale waves can be obtained (Haines and Holland 1998):

*ψ*

_{X→∞}(1/2

*X*)

^{X}

_{−X}

*ψ*

*dx*is the zonally averaged part of the streamfunction

*ψ*and

*L*

_{y}is the width of the beta-plane channel.

Separating (1) into planetary- and synoptic-scale equations has two advantages: (i) Equations (2a) and (2b) have a clear physical meaning in representing the interaction between an incipient block and synoptic-scale eddies compared to Eq. (1) and (ii) the analytical solution about the feedback of blocking development on the synoptic-scale eddies can be directly derived from Eq. (2b) using asymptotic expansion. Equation (2a) is the planetary-scale equation that is forced by the synoptic-scale disturbances, while (2b) is the planetary-to-synoptic-scale interaction equation that describes the feedback of developing block on the synoptic-scale eddies, which can be solved analytically. In (2a), *J*(*ψ*′, ∇^{2}*ψ*′)_{P} represents the contribution of synoptic-scale eddies to block development (Illari and Marshall 1983; Holopainen and Fortelius 1987), and is hereafter referred to as the eddy forcing or eddy feedback. As pointed out by Holopainen and Fortelius (1987) and Haines and Holland (1998), *J*(*ψ*′, ∇^{2}*ψ*′)_{P} plays a key role in the establishment of blocking, even though it is small compared to the other terms in (2a). In contrast, (2b) is linear in the synoptic-scale fields, but essentially nonlinear because the planetary-scale field *ψ* includes a contribution from synoptic- scale eddies *ψ*′ through Eq. (2a).

In this paper, the slowly varying scales are introduced to solve Eqs. (2a) and (2b), which are based upon the observational evidences presented by Holopainen and Fortelius (1987) and Lejenäs and Økland (1983). Holopainen and Fortelius showed that to a first-order approximation the planetary-scale flow in blocking cases behaves as a “free” stationary wave. This implies that the blocking wave can be described by the solution to the linear equation derived at the first (leading) order approximation of Eq. (2a). However, Lejenäs and Økland (1983) indicated that the blocking in most cases is a local phenomenon. This implies that the amplitude of the linear wave solution should be at least slowly varying so that the blocking flow can exhibit a local structure. Thus, it is appropriate to introduce the slowly varying scales in planetary-scale equation (2a). On the other hand, the synoptic-scale eddies should include the slowly varying scales because the feedback of block development plays a role in the deformation of synoptic-scale eddies.

### b. Asymptotic expansion

For our asymptotic analysis, it is useful to assume that *ψ* and *ψ*′ have *ψ* = ε*ψ̃**ψ*′ = ε^{3/2}*ψ̃**J*(*ψ*′, ∇^{2}*ψ*′)_{P} ∼ ε*J*(*ψ,* ∇^{2}*ψ*), obtained by Holopainen and Fortelius (1987) from the diagnostic study of a blocking event, where ε is an amplitude parameter much smaller than unity, whose square root is the ratio of the amplitudes of the synoptic-scale eddies and the planetary-scale waves. Such an assumption is useful for us to obtain a nonlinear wave amplitude equation with the eddy forcing. Of course, this restricts us to a particular asymptotic regime, but it is a regime of physical interest.

We choose *ψ*^{*}_{S}^{5/2}*ψ*′_{S}*h* = ε*h*′(*x,* *y*) is assumed in the quasigeostrophic regime (Pedlosky 1981).

*ψ̃*

*ψ̃*

Here we consider the special case that the amplitudes of the preexisting synoptic-scale eddies *ψ̃*′_{0}*X*_{1}, not on the scale *X*_{2} and slow time scales, in order to allow the preexisting synoptic-scale eddies to fix upstream of an incipient block. It must be pointed out that slow times, *T*_{1} and *T*_{2}, and slow spaces, *X*_{1} and *X*_{2}, in (5) reflect the first- and second-order modulation of the leading order Rossby wave *ψ̃*_{0}. Interestingly, *ψ̃*′_{1}*T*_{1}, *T*_{2}, *X*_{1}, and *X*_{2} due to the feedback of planetary-scale block flow *ψ̃*_{0}, even if *ψ̃*′_{0}

*ψ̃*′

_{0}

*ψ̃*′

_{1}

*ψ̃*′

_{0}

*ψ̃*′

_{0}

Equation (8a) is a linear Rossby wave equation with a possible topographic forcing. This equation reflects a fact noted by Holopainen and Fortelius (1987), who found in a case study of a blocking event, that to a first-order approximation the time-mean (planetary scale) blocking flow behaves as a free stationary wave. Thus, Eq. (8a) is reasonable as a first-order description of a blocking circulation. The amplitude evolution equation of slowly varying Rossby wave in (8a) can be derived from (8c).

### c. Derivation of the forced NLS equation

Frederiksen (1982) found that rapidly eastward-moving synoptic-scale disturbances possess a monopole cyclogenesis structure that is excited by three-dimensional instability. Thus, as a highly idealized case, the continuum spectrum of preexisting synoptic eddies is approximated, as in Luo (2000), by the sum of two Rossby waves having a monopole meridional structure and neighboring zonal wavenumbers, *k̃*_{1} and *k̃*_{2}, whose time scales are shorter than one week in order to assure that the eddy forcing *J*(*ψ̃*′_{0}^{2}*ψ̃*′_{0}_{P} has almost the same structure as the dipole component of blocking solution does. In this way, their nonlinear interaction projects onto the planetary-scale dipole wave.

*ψ̃*′

_{0}

*ψ̃*′

_{S}

*m*= −2

*π*/

*L*

_{y},

*ω̃*

_{1}=

*u*

_{0}

*k̃*

_{1}− (

*β*+

*Fu*

_{0})

*k̃*

_{1}/

*K̃*

^{2}

_{1}

*ω̃*

_{2}=

*u*

_{0}

*k̃*

_{2}− (

*β*+

*Fu*

_{0})

*k̃*

_{2}/

*K̃*

^{2}

_{2}

*K̃*

^{2}

_{n}

*k̃*

^{2}

_{n}

*m*

^{2}/4 +

*F,*

*k̃*

_{1}= (

*ñ*− Δ

*ñ*)

*k*

_{0},

*k̃*

_{2}= (

*ñ*+ Δ

*ñ*)

*k*

_{0}, Δ

*ñ*≪

*ñ,*

*X*

_{1}= ε

*x*;

*k*

_{0}= 1/[

*R*

_{0}cos(

*ϕ*

_{0})] is the zonal wavenumber of wave 1 around the earth at given latitude

*ϕ*

_{0}corresponding to a wavelength 2

*πR*

_{0}cos(

*ϕ*

_{0}),

*R*

_{0}is the nondimensional radius of the earth,

*ñ*is an integer, and c.c. represents complex conjugate of the preceding term. Here

*B*′

_{1}

*B*′

_{2}

*f*′

_{1}

*f*′

_{2}

*B*′

_{1}

*B*′

_{2}

*f*′

_{0}

*a*′

_{0}

*μ*(

*X*

_{1}+

*X*

_{0})

^{2}] if

*f*′

_{1}

*f*′

_{2}

*a*′

_{0}

*X*

_{0}= ε

*x*

_{o}represents the position of the maximum amplitude of preexisting eddies upstream, and

*μ*denotes the distribution of these eddies.

*h*

*h*

_{0}

*ikx*

*my*

*h*′

_{0}

*k*= 2

*k*

_{0}is the zonal wavenumber of the wavenumber two topography, and

*m*= −2

*π*/

*L*

_{y}.

Frederiksen (1982) found that the slowly moving growing large-scale wave associated with the onset of a blocking circulation exhibits a high-over-low dipole structure. Thus, it is natural to suppose that the first-order blocking solution described by (8a) is represented by the superposition of a topographically induced standing wave with a monopole meridional structure and a slowly varying stationary dipole Rossby wave linked to eddy forcing at third order.

*A*is the complex amplitude of dipole Rossby wave whose amplitude is slowly varying in the zonal direction, while the second term on the right-hand side of (11) represents a topographically induced standing monopole wave. It is clear that, when the free dipole Rossby wave tends to be stationary (

*ω*= 0), the topographically induced resonance proposed by Tung and Lindzen (1979) cannot be allowed in such a background flow. When the topography vanishes, (11) represents a dipole blocking solution.

The topography can play a large role in the amplification of monopole wave if *u*_{0} → *β*/(*k*^{2} + *m*^{2}/4). This corresponds to the resonant theory proposed by Tung and Lindzen (1979). However, the strong background westerly wind will exclude the stationary dipole wave [*u*_{0} = *β*/(*k*^{2} + *m*^{2}) for *ω* = 0] as a precondition of block onset. In this case, the role of synoptic-scale eddies will becomes unimportant. As pointed out by Shutts (1983), blocking does not occur for more rapid flows in which a stationary wave cannot be excited. In this paper, we consider a stationary dipole Rossby wave that allows *u*_{0} = *β*/(*k*^{2} + *m*^{2}) (≈7 m s^{−1} in the dimensional form) for *ω* = 0. This condition is easily satisfied for the background environments over the Atlantic and Pacific Oceans.

*ψ̃*

_{1}

*ψ*

_{2}

*y,*

*T*

_{1}

*T*

_{2}

*X*

_{1}

*X*

_{2}

*ψ*

_{2}and its solution can be obtained as

*A*|

^{2}=

*AA**,

*A** is the complex conjugate of

*A,*and the coefficients in (14) are given in the appendix. Note that the solutions derived above satisfy the boundary conditions automatically.

Choosing *L* = 10^{6} m, *U* = 10 m s^{−1}, *ϕ*_{0} = 55°N, *L*_{y} = 5 (5000 km in dimensional form), *F* = 1, *u*_{0} = *β*/(*k*^{2} + *m*^{2}), *ñ* = 10 and Δ*ñ* = 0.75, both |*k* − (*k̃*_{2} − *k̃*_{1})| ≪ 1 and *ω̃*_{2} − *ω̃*_{1} − *ω* ≪ 1 hold. In this case, the near resonance between the free dipole Rossby wave for zonal wavenumber 2 and the eddy forcing is allowed (Franzke et al. 2000; Luo 2000).

*k*− (

*k̃*

_{2}−

*k̃*

_{1}) = εΔ

*K*and

*ω̃*

_{2}−

*ω̃*

_{1}−

*ω*= εΩ(

*ω*= 0 in this paper) and inserting (9), (10), (11), (12), and (14) into (8c), a forced NLS equation can be obtained as

Equation (17) is a forced NLS equation, which describes the amplification of dipole wave in an incipient block through upscale energy transfer under the influence of topography even though it does not explicitly contain the topography term. If (17) has a soliton solution, then the solution (11) represents a local blocking.

## 3. An analytical solution of the feedback of block evolution on preexisting synoptic-scale eddies

Note that (18) is the second-order solution to the preexisting synoptic-scale eddies associated with block onset, which crudely represents the feedback of the block formation on the preexisting synoptic-scale eddies. The change in the synoptic-scale field can be predicted by (18) if the block evolution is known from (17). It is easy to see from (18) that the change in the synoptic-scale eddies associated with the onset of a blocking is attributable to two parts: the developing blocking mode (*ψ*′_{B}*ψ*′_{T}*ψ*′_{B}*A* of the dipole wave and the synoptic-scale eddy wavenumber and amplitude, *ψ*′_{B}*ψ̃*′_{1}*ψ̃*′_{0}*ψ̃*′_{1}*ψ̃*′_{0}

## 4. Numerical results on the interaction between an incipient block and preexisting synoptic-scale eddies over a wavenumber-2 topography during the life cycle of blocking

It is difficult to solve (17) analytically, but a finite-difference method can be applied (Taha and Ablowitz 1984). If the solutions to (17) are obtained, the different role of the synoptic-scale eddies and topography in exciting a blocking anticyclone can be clarified. In fact, the analytical solution to (17) can be derived by the perturbed inverse scattering method (Hasegawa and Kodama 1995). In this paper, we present a numerical solution of (17).

The finite-difference method of Taha and Ablowitz (1984) is used to solve (17) for given initial conditions. The model is run for 20 000 time steps with time and space grid intervals of Δ*T* = 0.0005 and Δ*Z* = 0.08, respectively, over the space domain of −20 ≤ *Z* ≤ 20. The total time of the model run is *T* = 2 × 10^{4}Δ*T* = 10 (near 200 days in dimensional form). Periodic boundary conditions are used in the zonal direction.

### a. Incipient diffluent flow and oscillation of eddy-forced soliton

In this paper, the atmospheric parameters *f*_{0} = 10^{−4} s^{−1}, *D* = 10^{4} m, *L* = 10^{6} m, *U* = 10 m s^{−1}, and *h**L*_{y} = 5, *F* = 1.0, and *ϕ*_{0} = 55°N are chosen as an example. It is easy to find *λδ* > 0 in (17) for zonal wavenumber 2 in the mid–high latitude region in a wide parameter range. Thus, the small-amplitude envelope soliton solution to (17) without forcing can be approximately regarded as an incipient block because observed blocks are usually a local phenomenon (Lejenäs and Økland 1983).

Figure 2 shows the horizontal distribution of wavenumber-2 topography (*h*) in the region of −5.6 ≤ *x* ≤ 5.6 corresponding to one nondimensional wavelength and *h*′_{0}

*A*

_{0}is the value of

*A*(

*Z,*0) at

*Z*= 0. Note that the value of

*A*

_{0}represents the strength of an antecedent planetary-scale soliton.

It is useful to define *M*(*T*) = |*A*(*Z,* *T*)|_{max} as the maximum amplitude of the forced envelope soliton. If we allow the parameters *A*_{0} = 0.35/ε, *h*′_{0}*μ* = 2.4, ε = 0.24, and *X*_{0} = 2.87ε/2, then the time evolution of *M*(*T*) is shown in Fig. 3 for *a*′_{0}^{3/2}.

It can be found in Fig. 3 that the small-amplitude envelope Rossby soliton amplifies periodically through upscale eddy energy transfer and exhibits an oscillation with a period of *T* = 0.95 (day 19). Throughout the entire process the envelope soliton preserves its soliton-like state. The amplifying of the incipient soliton in Fig. 3 corresponds to the onset of blocking, and the decay of the soliton amplitude corresponds to the decay of blocking. Thus, it can be concluded that the synoptic-scale waves either reinforce or destroy the block (Anderson 1995; Cash and Lee 2000). Although the topography appears to be unimportant for the amplification of the dipole component of an incipient block compared to the role of preexisting synoptic-scale eddies because no topographic terms are included in (17), but it seems to play an important role in the block onset and the split of synoptic-scale eddies.

### b. Block onset associated with synoptic-scale eddies and controlling factors of northward deflection of the storm track

To examine the different roles of synoptic-scale eddies and topography in block onset and how the deformation of synoptic-scale eddies is affected by the topography, we will use (11) and (18) to present the interaction between an incipient block and preexisting synoptic-scale eddies with and without topography by solving (17). Here, we choose the same parameter values as in Fig. 3 as an example. For *h*′_{0}*h*′_{0}*ψ*_{e} = −*u*_{0}*y* + *ψ*), synoptic-scale field (*ψ*′), and the total field (Ψ_{T} = *ψ*_{e} + *ψ*′) in the interaction between an incipient block and preexisting synoptic-scale eddies, are shown in Figs. 4 (without topography) and 5 (with topography), respectively.

Figure 4a shows that the incipient envelope soliton exhibits a weak vortex pair, which is seen as a highly idealized, incipient dipole block prior to block onset. The prespecified (antecedent) eddies are located upstream of the incipient block, as shown in Fig. 4b for day 0. The total field in Fig. 4c shows that there exist small-scale troughs and ridges upstream of the incipient block centered at *x* = 0. These small-scale troughs and ridges are the cyclone waves that play a central role in the block development (Berggren et al. 1949; Colucci 1985; Lupo and Smith 1995). At the same time, the synoptic eddies deform due to the feedback of subsequent block onset. The subsequent change of preexisting synoptic eddies and the incipient block is easily seen from Figs. 4a and 4b. The block evolution from an incipient block into a typical dipole block can be explained in terms of the time-dependent eddy-forcing term *J*_{P} = −*J*(*ψ̃*′_{0}^{2}*ψ̃*′_{0}_{P} induced by the prespecified transient eddies (*ψ̃*′_{0}*A*(*Z,* 0) = 0.35/ε] or the other function is chosen as an initial value of (17) (not shown).

Figure 4c indicates that before day 9, the anticyclonic ridges tend to be enhanced and drift northward, but the cyclonic troughs tend to deepen and drift southward due to the role of the eddy forcing upstream. This process will lead to the troughs and ridges becoming cut off from the main current occasionally forming isolated cyclonic vortices to the south and isolated anticyclonic vortices to the north. This strengthens the meander of the blocking flow and bears a resemblance to a Berggren-type blocking flow first observed by Berggren et al. 1949, their Fig. 26), where there are several isolated anticyclonic or cyclonic vortices within the blocking region. The Berggren-type blocking is an indicator of the central role played by the synoptic-scale eddies. It is suggested that what type of forcing in exciting blocking could be found as long as the blocking pattern is observed in detail. For example, the synoptic eddies are considered to play an important role in the block onset if observed blocking is found to be composed of several isolated anticyclonic or cyclonic vortices (Butchart et al. 1989). Therefore, the planetary-scale projection of self-interaction among synoptic-scale eddies (eddy forcing) may be the most important contributor to the amplification and decay of meander blocking. Shutts (1983) suggested that the transfer of eddy energy to the larger-scale split-jet flow field is achieved by the action of the basic-state deformation field on the synoptic eddies. In fact, not all blocks possess such a process. As indicated by Colucci (1985), the block onset appears to be caused mainly by the interaction between antecedent cyclone waves and a planetary-scale ridge. In this interaction, the transfer of eddy energy to the block dipole is likely to be determined by the pattern of time-dependent eddy forcing (*J*_{P}), but not by the planetary-scale deformation field (*ψ̃*_{0}) that modulates the synoptic-scale eddies. It is further found from Fig. 4b that the synoptic eddies are enhanced and split into two symmetric branches around the blocking region during the block development, which is in agreement with the observed change of synoptic-scale eddies during the life cycle of blocking (Nakamura and Wallace 1993). The reason for enhanced synoptic-scale activity is easily explained in terms of (18). Because the induced eddies involve the amplitude (*A*), the synoptic-scale eddy activity will be enhanced if the incipient block is intensified. Thus, it is natural that the activity of observed synoptic-scale eddies is enhanced during block onset (Higgins and Schubert 1994). The direct numerical solutions of barotropic and two-layer quasigeostrophic potential vorticity equations by Shutts (1983), Vautard et al. (1988), and Vautard and Legras (1988) also confirmed the analytical results obtained here.

Figure 5 shows the interaction of an incipient blocking ridge over the central part of the ocean (topographic trough for *h*′_{0}

Figure 5a shows the planetary-scale evolution of a blocking high with topography, which is standing and extremely different from that in Fig. 4a where a dipole blocking only occurs and is slowly westward traveling, even if *ω* = 0 for the dipole wave is satisfied. This suggests that the prevalence of a standing blocking high circulation may be due to topography acting together with synoptic-scale eddies, which is in close agreement with the observational result of Higgins and Schubert (1994). The modulated synoptic-scale eddies shown in Fig. 5b are still split into two branches around the blocking region, but the northern branch is much stronger than the southern one. This figure replicates a realistic evolution of synoptic-scale activity that exhibits a northward shift during the life cycle of blocking anticyclones observed over the North Pacific by Higgins and Schubert (1994). A prerequisite condition for the noticeable northward deflection of synoptic-scale eddies associated with blocking circulation is that the topographically induced standing ridge must be almost in phase with the dipole component of the blocking flow. Otherwise, no strong blocking high occurs if the topographically induced standing ridge has a large phase mismatch relative to the dipole soliton (not shown).

In addition, it is found that the planetary-scale part does look like real blocking (Figs. 4a and 5a), but when the synoptic scales are added, the total field seems to exhibit a meandering blocking (Figs. 4c and 5c), which is in agreement with that observed by Berggren et al. (1949, their Figs. 14–18). This kind of blocking structure is usually observed in the real atmosphere (Berggren et al. 1949; Butchart et al. 1989). Omega or dipolar-shaped blocking is only observed when the synoptic scales are filtered out (Higgins and Schubert 1994). During the process of block onset the synoptic-scale eddies propagating eastward will become incorporated into the block region to finally form a meandering block, which has been supported by the observational study by Nakamura and Wallace (1993) and the numerical experiment by Haines and Marshall (1987). In real blocking processes, the meandering of blocking is also influenced by several other factors such as the strength of the background westerly wind, the intensities and positions of preexisting synoptic-scale eddies, the strength of the incipient block, and so on. For example, the meandering of blocking will become weak if the strength of the preexisting block is decreased. This point can be easily confirmed by some simple calculations. On the other hand, it can be found that the result obtained here does not depend on the choice of small parameter ε (not shown).

### c. Role of topography

In this subsection, we will discuss what role the topography effect plays in the deformation of preexisting synoptic-scale eddies during block onset from (18c). For the same parameters as in Fig. 5, Fig. 6 shows the time dependent field of *ψ*′_{T}

It is easy to see from Fig. 6 that the interaction between the wavenumber-2 topography and preexisting synoptic-scale eddies with monopole meridional structure tends to induce new synoptic-scale eddies having a dipole meridional structure. These eddies are time dependent. It is found from Fig. 4b that the synoptic-scale eddies without topography always exhibit a symmetric meridional change due to feedback of the developing block. Comparing Fig. 6 with Fig. 4b shows that the topographically induced eddies tend to be in phase with the northern eddies and out of phase with the southern eddies apart from on day 0. This tendency will strengthen the synoptic-scale eddies over the northern side of the channel and weaken the southern eddies so that the storm track organized by the modulated synoptic-scale eddies deflects northward. This may explain why the northern branch of the storm track is dominant during blocking episodes (Higgins and Schubert 1994). However, if the topography is 180° out of phase with the one considered here, the topographically induced standing tough in phase with the dipole soliton will deflect storms southward (not shown). But this case does not correspond to a blocking event, as shown in Fig. 1. Thus, it can be concluded that the topographically induced planetary-scale standing ridge that is in phase with the dipole component of blocking flow is an important controlling factor for the northward deflection of storm tracks associated with block onset. Furthermore, it must be pointed out that the result obtained here does not strongly depend upon the initial structure of a dipole wave in an incipient block (not shown).

### d. Role of planetary–synoptic-scale interaction

In this subsection, we will discuss the contribution of synoptic–planetary-scale interaction terms *J*(*ψ*′, ∇^{2}*ψ* + *h*) + *J*(*ψ,* ∇^{2}*ψ*′) to block onset. Figure 7 shows the total field during the life cycle of a blocking anticyclone in the absence of synoptic–planetary-scale interaction terms [*J*(*ψ*′, ∇^{2}*ψ* + *h*) + *J*(*ψ,* ∇^{2}*ψ*′) = 0, which corresponds to *ψ̃*′_{1}

It is clear that, in the absence of synoptic–planetary-scale interaction, Fig. 7 does not exhibit many troughs and ridges within the blocking region. This indicates that the synoptic–planetary-scale interaction controls the deformation of the synoptic-scale field related to the life cycle of a blocking. This point can be further verified by calculating the streamfunction field of synoptic-scale eddies without including the synoptic–planetary-scale interaction (not shown). In fact, the synoptic–planetary-scale interaction reflects the feedback of excited blocking on the preexisting eddies. If this feedback is ignored, no realistic blocking flow with a strong meander similar to that first found by Berggren et al. (1949) can be detected. In contrast, if the planetary–synoptic-scale interaction is included, one not only finds a realistic blocking circulation (Fig. 4c), but also a realistic change in the modulated synoptic-scale eddies (Fig. 4b). Although Figs. 4c and 5c are highly idealized, this kind of blocking structure is usually observed in the real atmosphere (Luo et al. 2001, their Fig. 1). Thus, the synoptic–planetary-scale interaction is a most important contributor to the change of synoptic-scale eddies associated with the onset and decay of blocking and is a controlling factor for the meandering of blocking flow.

## 5. Conclusions and discussion

In this paper, a new theoretical model is proposed to examine the interaction between a stationary planetary-scale incipient block and synoptic-scale eddies based upon the concept of an envelope Rossby soliton and an analytic solution of the feedback of block development on preexisting synoptic-scale eddies. Using this asymptotic model the different roles of synoptic-scale eddies and wavenumber-2 topography in the onset, maintenance, and decay of blocking are identified. The main conclusions are as follows:

Planetary-scale projection of self-interaction among synoptic-scale eddies plays a key role in block onset compared to the forcing of large-scale land–sea contrast topography (idealized as wavenumber-2 topography). The synoptic–planetary-scale interaction processes appear only to be important for the subsequent deformation of preexisting synoptic-scale eddies related to block onset, which causes the meandering of the blocking flow. The change in the modulated synoptic-scale eddies on time scale tends to be consistent with the block evolution. The feedback from a localized block is able to localize eddy activity.

The storm track eddies can grow more rapidly over the northwest side of the blocking region than over the southwest side due to the role of a topographically induced planetary-scale standing ridge that is almost in phase with the stationary dipole wave in the blocking region, inducing a northward deflection of storm tracks around the blocking region.

In the absence of large-scale topography, the synoptic-scale eddies tend to make the formed block shift westward, but the topography tends to make the blocking circulation standing and to fix its geographical position.

It should be pointed out that, even though the basic idea that the blocking dipole is resonantly forced by eddies is similar to the previous studies by Shutts (1983) and Haines and Marshall (1987), the dynamical framework is very different. In Shutts (1983), the block dipole was assumed to be time independent. In Haines and Marshall (1987), the nondispersive modon that corresponds to the mature dipole block was chosen as a prototype of a block dipole. These assumptions preclude the dispersive and nondispersive states of the blocking phenomenon at different phases of block evolution (Luo 2000). Our model here is able to avoid such a restriction.

Furthermore, we have shown theoretically that the geographical position for the formation of the coherent block structure can be fixed by the phase of the long wave that is forced directly by the topography. Probably, the blocking pattern would become slightly different if many topographic modes were considered, but the basic results would be similar. The results obtained in the present paper are of significance for understanding why an omega-type block is more prominent over the Pacific than over the Atlantic. This possible cause may be that a large-scale topographic ridge is so easily excited over the Pacific that the monopole component of blocking flow is large. In addition, the present study does not consider the role of all synoptic-scale waves. As pointed out by Young and Villere (1985), there can be significant cancellation in the eddy forcing when all synoptic-scale waves are present. In addition, if there is a traveling planetary-scale monopole wave in the incipient block prior to block onset, how this traveling wave affects the evolution of the blocking anomaly and synoptic-scale eddy activity is unclear. This issue will be reported in Part II (Luo 2005).

## Acknowledgments

This work was carried out while the author was a visiting scientist at the Department of Physics, University of Toronto, supported by the distinguished visiting scholar program of the Chinese government and the National Natural Science Foundation of China (40175011) and the outstanding Youth Natural Science Foundation of China (40325016). The author would like to thank Prof. T. G. Shepherd for providing pleasant working conditions during his visit and many useful suggestions that, led to much improvement of the original manuscript. Drs. Lucy Campbell and Shuzhan Ren and Mr. Mark Fruman are also acknowledged for their useful discussions and assistance. The author also thanks two anonymous reviewers for useful suggestions in improving this paper.

## REFERENCES

Anderson, J. L., 1995: A simulation of atmospheric blocking with a forced barotropic model.

,*J. Atmos. Sci***52****,**2593–2607.Berggren, R., B. Bolin, and C. G. Rossby, 1949: An aerological study of zonal motion, its perturbations and break-down.

,*Tellus***1****,**14–37.Butchart, N., K. Haines, and J. C. Marshall, 1989: A theoretical study of solitary waves and atmospheric blocking.

,*J. Atmos. Sci***46****,**2063–2078.Cai, M., and H. M. Van den Dool, 1991: Low-frequency waves and traveling storm tracks. Part I: Barotropic component.

,*J. Atmos. Sci***48****,**1420–1436.Cash, B. A., and S. Lee, 2000: Dynamical processes of block evolution.

,*J. Atmos. Sci***57****,**3202–3218.Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking.

,*J. Atmos. Sci***36****,**1205–1216.Colucci, S. J., 1985: Explosive cyclogenesis and large-scale circulation changes: Implications for atmospheric blocking.

,*J. Atmos. Sci***42****,**2701–2717.Egger, J., 1978: Dynamics of blocking high.

,*J. Atmos. Sci***35****,**1788–1801.Franzke, C., K. Fraedrich, and F. Lunkeit, 2000: Low-frequency variability in a simplified atmospheric global circulation model: Storm-track induced “spatial resonance.”.

,*Quart. J. Roy. Meteor. Soc***126****,**1691–2708.Frederiksen, J. S., 1982: A unified three-dimensional instability theory of the onset of blocking and cyclogenesis.

,*J. Atmos. Sci***39****,**969–987.Green, J. S. A., 1977: The weather during July 1977: Some dynamical consideration of the drought.

,*Weather***32****,**120–126.Haines, K., and J. C. Marshall, 1987: Eddy-forced coherent structures as a prototype of atmospheric blocking.

,*Quart. J. Roy. Meteor. Soc***113****,**681–704.Haines, K., and A. J. Holland, 1998: Vacillation cycles and blocking in a channel.

,*Quart. J. Roy. Meteor. Soc***124****,**873–895.Hasegawa, A., and Y. Kodama, 1995:

*Solitons in Optical Communications*. Clarendon Press, 320 pp.Higgins, R. W., and S. D. Schubert, 1994: Simulated life cycles of persistent anticyclonic anomalies over the North Pacific: Role of synoptic-scale eddies.

,*J. Atmos. Sci***51****,**3238–3259.Holopainen, E., and C. Fortelius, 1987: High-frequency transient eddies and blocking.

,*J. Atmos. Sci***44****,**1632–1645.Illari, L., and J. C. Marshall, 1983: On the interpretation of eddy fluxes during a blocking episode.

,*J. Atmos. Sci***40****,**2232–2242.Jeffrey, A., and T. Kawahara, 1982:

*Asymptotic Methods in Nonlinear Wave Theory*. Pitman Press, 273 pp.Juckes, M. N., and M. E. McIntyre, 1987: A high-resolution one-layer model of breaking planetary waves in the stratosphere.

,*Nature***328****,**590–596.Lejenäs, H., and H. Økland, 1983: Characteristics of Northern Hemisphere blocking as determined from a long time series of observed data.

,*Tellus***35A****,**350–362.Lejenäs, H., and R. A. Madden, 1992: Traveling planetary-scale waves and blocking.

,*Mon. Wea. Rev***120****,**2821–2830.Luo, D., 2000: Planetary-scale baroclinic envelope Rossby solitons in a two-layer model and their interaction with synoptic-scale eddies.

,*Dyn. Atmos. Oceans***32****,**27–74.Luo, D., 2005: A barotropic envelope Rossby soliton model for block– eddy interaction. Part II: Role of westward-traveling planetary waves.

*J. Atmos. Sci.,***62,**22–40.Luo, D., F. Huang, and Y. Diao, 2001: Interaction between antecedent planetary-scale envelope soliton blocking anticyclone and synoptic-scale eddies: Observations and theory.

,*J. Geophys. Res***106****,**31795–31816.Lupo, A. R., and P. J. Smith, 1995: Planetary and synoptic-scale interactions during the life cycle of a mid-latitude blocking anticyclone over the North Atlantic.

,*Tellus***47A****,**575–596.Malanotte-Rizzoli, P., and P. Malguzzi, 1987: Coherent structures in a baroclinic atmosphere. Part III: Block formation and eddy forcing.

,*J. Atmos. Sci***44****,**2493–2505.Malguzzi, P., 1993: An analytical study on the feedback between large- and small-scale eddies.

,*J. Atmos. Sci***50****,**1429–1436.McIntyre, M. E., and T. N. Palmer, 1983: Breaking planetary waves in the stratosphere.

,*Nature***305****,**593–600.Nakamura, H., and J. M. Wallace, 1993: Synoptic behavior of baroclinic eddies during the blocking onset.

,*Mon. Wea. Rev***121****,**1892–1903.Orlanski, I., 1998: Poleward deflection of storm tracks.

,*J. Atmos. Sci***55****,**2577–2602.Pedlosky, J., 1981: Resonant topographic waves in barotropic and baroclinic flows.

,*J. Atmos. Sci***38****,**2626–2641.Shepherd, T. G., 1987: A spectral view of nonlinear fluxes and stationary–transient interaction in the atmosphere.

,*J. Atmos. Sci***44****,**1166–1178.Shutts, G. J., 1983: The propagation of eddies in diffluent jetstreams: Eddy vorticity forcing of blocking flow fields.

,*Quart. J. Roy. Meteor. Soc***109****,**737–761.Stewartson, K., 1978: The evolution of the critical layer of a Rossby wave.

,*Geophys. Astrophys. Fluid Dyn***9****,**185–200.Taha, T. R., and M. J. Ablowitz, 1984: Analytical and numerical aspects of certain nonlinear evolutions. Part II: Numerical nonlinear Schrödinger equation.

,*J. Comput. Phys***55****,**203–230.Tsou, C. S., and P. J. Smith, 1990: The role of synoptic/planetary-scale interactions during the development of a blocking anticyclone.

,*Tellus***42A****,**174–193.Tung, K. K., and R. S. Lindzen, 1979: A theory of stationary long waves. Part I: A simple theory of blocking.

,*Mon. Wea. Rev***107****,**714–734.Vautard, R., and B. Legras, 1988: On the source of midlatitude low-frequency variability. Part II: Nonlinear equilibration of weather regimes.

,*J. Atmos. Sci***45****,**2845–2867.Vautard, R., B. Legras, and M. Deque, 1988: On the source of midlatitude low-frequency variability. Part I: A statistical approach to persistence.

,*J. Atmos. Sci***45****,**2811–2843.Warn, T., and H. Warn, 1978: The evolution of a nonlinear critical layer.

,*Stud. Appl. Math***59****,**37–71.Warn, T., and B. Brasnett, 1983: The amplification and capture of atmospheric solitons by topography: A theory of the onset of regional blocking.

,*J. Atmos. Sci***40****,**28–38.Young, R. E., and G. L. Villere, 1985: Nonlinear forcing of planetary scale waves by amplifying unstable baroclinic eddies generated in the troposphere.

,*J. Atmos. Sci***42****,**1991–2006.

## APPENDIX

Horizontal distribution of wavenumber-2 topography. Contour interval (CI) is 0.1. Solid contours are positive and dashed contours are negative

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Horizontal distribution of wavenumber-2 topography. Contour interval (CI) is 0.1. Solid contours are positive and dashed contours are negative

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Horizontal distribution of wavenumber-2 topography. Contour interval (CI) is 0.1. Solid contours are positive and dashed contours are negative

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Time evolution of *M*(*T*) = |*A*(*Z,* *T*)|_{max} for the parameters *A*_{0} = 0.35/ε, *h*′_{0}*μ* = 2.4, *a*′_{0}^{3/2}, *X*_{0} = 2.87ε/2, and ε = 0.24

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Time evolution of *M*(*T*) = |*A*(*Z,* *T*)|_{max} for the parameters *A*_{0} = 0.35/ε, *h*′_{0}*μ* = 2.4, *a*′_{0}^{3/2}, *X*_{0} = 2.87ε/2, and ε = 0.24

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Time evolution of *M*(*T*) = |*A*(*Z,* *T*)|_{max} for the parameters *A*_{0} = 0.35/ε, *h*′_{0}*μ* = 2.4, *a*′_{0}^{3/2}, *X*_{0} = 2.87ε/2, and ε = 0.24

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Interaction of an initial envelope Rossby soliton (an incipient block) with preexisting synoptic-scale eddies without topography effect for the parameters *A*_{0} = 0.35/ε, *a*′_{0}^{3/2}, *μ* = 2.4, *X*_{0} = 2.87ε/2, and ε = 0.24: (a) planetary-scale field, CI = 0.15; (b) synoptic-scale field in which the solid contours are positive and the dashed contours are negative, CI = 0.3; and (c) total field, CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Interaction of an initial envelope Rossby soliton (an incipient block) with preexisting synoptic-scale eddies without topography effect for the parameters *A*_{0} = 0.35/ε, *a*′_{0}^{3/2}, *μ* = 2.4, *X*_{0} = 2.87ε/2, and ε = 0.24: (a) planetary-scale field, CI = 0.15; (b) synoptic-scale field in which the solid contours are positive and the dashed contours are negative, CI = 0.3; and (c) total field, CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Interaction of an initial envelope Rossby soliton (an incipient block) with preexisting synoptic-scale eddies without topography effect for the parameters *A*_{0} = 0.35/ε, *a*′_{0}^{3/2}, *μ* = 2.4, *X*_{0} = 2.87ε/2, and ε = 0.24: (a) planetary-scale field, CI = 0.15; (b) synoptic-scale field in which the solid contours are positive and the dashed contours are negative, CI = 0.3; and (c) total field, CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 4 but with topography (*h*′_{0}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 4 but with topography (*h*′_{0}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 4 but with topography (*h*′_{0}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Instantaneous fields (*ψ*′_{T}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Instantaneous fields (*ψ*′_{T}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

Instantaneous fields (*ψ*′_{T}

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 5, but for the case without planetary–synoptic-scale interaction terms; CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 5, but for the case without planetary–synoptic-scale interaction terms; CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1

As in Fig. 5, but for the case without planetary–synoptic-scale interaction terms; CI = 0.3

Citation: Journal of the Atmospheric Sciences 62, 1; 10.1175/1186.1