a. Model description

The nondimensional, equivalent barotropic vorticity equation with large-scale bottom topography and an external vorticity source but without dissipation can be written as (Egger 1978)

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where Ψ_T is the streamfunction, F = (L/R_d)², and β = β₀L²/U; here R_d is the Rossby deformation radius, β₀ is the meridional gradient of the Coriolis parameter;

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is the nondimensional topography. The characteristic scales of the horizontal length, velocity, and topographic height are 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; f₀ is the fixed Coriolis parameter; D is the mean depth of the fluid; and ψ^*_S is a time-dependent synoptic-scale vorticity source term.

The diagnostic study of blocking case by Holopainen and Fortelius (1987) indicates that the zonal scales of blocking wave and synoptic-scale eddies are remarkably different. In this case, it is appropriate to separate Eq. (1) into planetary and synoptic scales. We split the streamfunction Ψ_T into three parts: Ψ_T = ψ₀ + ψ + ψ′, where ψ₀ = −u₀y is the basic-state field (where u₀ 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, Δk̃, satisfies Δk̃ ∼ k ≪ k̃_n. These assumptions are idealized, but are motivated by observations (Shepherd 1987). In this case, the self-interaction term J(ψ, ∇²ψ) of the planetary wave lies within the planetary-scale range, while the planetary–synoptic-scale interaction terms J(ψ′, ∇²ψ + h) and J(ψ, ∇²ψ′) 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′, ∇²ψ + h) and J(ψ, ∇²ψ′) are approximately in the synoptic-scale range. Furthermore, the synoptic-scale interaction term, denoted by J(ψ′, ∇²ψ′)_P, is a projection onto the planetary scale, while the projection of the synoptic-scale component, denoted by J(ψ′, ∇²ψ′)_S, onto the synoptic scales near k̃_n is negligible because it has zonal wavenumber k̃₁ + k̃₂(k̃₁ + k̃₂ ≫ k̃_n). Thus, J(ψ′, ∇²ψ′)_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):

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The lateral boundary conditions at the two rigid walls are (Pedlosky 1981)

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where ψ = lim_X→∞ (1/2X) ∫^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′, ∇²ψ′)_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′, ∇²ψ′)_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 = εψ̃ and ψ′ = ε^3/2ψ̃′ based upon an observational fact of J(ψ′, ∇²ψ′)_P ∼ εJ(ψ, ∇²ψ), 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 = ε^5/2ψ′_S so that the slow space-varying amplitudes of the preexisting synoptic-scale eddies, as defined below, are balanced by a prescribed synoptic-scale vorticity source. Finally, the bottom topography is considered to be wavy and h = εh′(x, y) is assumed in the quasigeostrophic regime (Pedlosky 1981).

With these assumptions, Equations (2a) and (2b) can be rewritten as

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The evolution of quasi-stationary planetary-scale waves during the life cycle of blocking can be understood as the slow modulation of the amplitude of the leading-order planetary-scale wave associated with the synoptic-scale eddies. The nonlinear modulated wave is generally described by a nonlinear Schrödinger (NLS) equation (Jeffrey and Kawahara 1982). The commonly used approach is to introduce slow space and time scales (variables). To obtain the equation governing the amplitude modulation of a Rossby wave, it is necessary to introduce the following slow time and space variables:

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We expand the fields ψ̃ and ψ̃′ as

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Here we consider the special case that the amplitudes of the preexisting synoptic-scale eddies ψ̃′₀ can be considered to have a variation on the scale X₁, not on the scale X₂ 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, T₁ and T₂, and slow spaces, X₁ and X₂, in (5) reflect the first- and second-order modulation of the leading order Rossby wave ψ̃₀. Interestingly, ψ̃′₁ inevitably contains slow time and spatial variables T₁, T₂, X₁, and X₂ due to the feedback of planetary-scale block flow ψ̃₀, even if ψ̃′₀ can be assumed to be independent of slow variables.

Using (5) and (6a) and (6b), (4b) yields the first two orders in ε:

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It is clear that to first-order approximation, the preexisting synoptic-scale eddies (ψ̃′₀) are linear free Rossby waves, but the second-order solution (ψ̃′₁) in (7b) is induced by the feedback of block development and is a modification to the preexisting synoptic-scale eddies (ψ̃′₀) if

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is allowed to eliminate the secular terms in (7b). Based on this treatment, the interaction between an incipient block and preexisting synoptic-scale eddies can be investigated. As demonstrated by Colucci (1985), the onset of blocking can be due to the anomalous planetary-scale advection of synoptic-scale potential vorticity associated with the antecedent cyclones through the planetary–synoptic scale interaction. In the present paper, the preexisting synoptic-scale eddies (ψ̃′₀) prescribed here correspond to the antecedent cyclones.

The following equations corresponding to the first three orders in ε can be obtained by substituting (5) and (6a)–(6b) into (4a)

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where the boundary conditions for (8a)–(8c) are also obtained by substituting (6a) and (6b) into (3).

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, k̃₁ and k̃₂, whose time scales are shorter than one week in order to assure that the eddy forcing J(ψ̃′₀, ∇²ψ̃′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.

The preexisting synoptic-scale eddies (ψ̃′₀) and synoptic-scale vorticity source (ψ̃′_S) prescribed in this paper are thus assumed to be

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where m = −2π/L_y, ω̃₁ = u₀k̃₁ − (β + Fu₀)k̃₁/K̃²₁, ω̃₂ = u₀k̃₂ − (β + Fu₀)k̃₂/K̃²₂, K̃²_n = k̃²_n + m²/4 + F, k̃₁ = (ñ − Δñ)k₀, k̃₂ = (ñ + Δñ)k₀, Δñ ≪ ñ, X₁ = εx; k₀ = 1/[R₀ cos(ϕ₀)] is the zonal wavenumber of wave 1 around the earth at given latitude ϕ₀ corresponding to a wavelength 2πR₀ cos(ϕ₀), R₀ is the nondimensional radius of the earth, ñ is an integer, and c.c. represents complex conjugate of the preceding term. Here B′₁ and B′₂ represent the real amplitudes of the preexisting synoptic-scale eddies whose magnitudes are determined by the amplitudes f ′₁ and f ′₂ of the prescribed synoptic-scale vorticity source.

We can obtain B′₁ = B′₂ = f ′₀ = a′₀ exp[−μ(X₁ + X₀)²] if f ′₁ and f ′₂ are prescribed by

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from the nonsecular term equation derived in section 2b, where a′₀ is the amplitude of the preexisting synoptic-scale eddies, X₀ = εx_o represents the position of the maximum amplitude of preexisting eddies upstream, and μ denotes the distribution of these eddies.

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.

It is thus natural to seek a solution to (8a) of the following form:

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where

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and 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 u₀ → β/(k² + m²/4). This corresponds to the resonant theory proposed by Tung and Lindzen (1979). However, the strong background westerly wind will exclude the stationary dipole wave [u₀ = β/(k² + m²) 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 u₀ = β/(k² + m²) (≈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.

If a zonal averaging is made after substituting (11) into (8b), using the solvability condition for (8b) we can obtain

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where

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Obviously, a special solution to (12b) is

ψ̃₁ψ₂y,T₁T₂X₁X₂(13)

A zonally averaging of (8c) yields an equation for ψ₂ and its solution can be obtained as

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where |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 = 10⁶ m, U = 10 m s⁻¹, ϕ₀ = 55°N, L_y = 5 (5000 km in dimensional form), F = 1, u₀ = β/(k² + m²), ñ = 10 and Δñ = 0.75, both |k − (k̃₂ − k̃₁)| ≪ 1 and ω̃₂ − ω̃₁ − 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).

Defining k − (k̃₂ − k̃₁) = εΔK and ω̃₂ − ω̃₁ − 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

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where the coefficients are given in the appendix.

If one introduces the coordinates

ZX₁C_gT₁TT₁(16)

then Eq. (15) can be rewritten as

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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.

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

In this paper, the atmospheric parameters f₀ = 10⁻⁴ s⁻¹, D = 10⁴ m, L = 10⁶ m, U = 10 m s⁻¹, and h = 1000 m, ε = 0.24, L_y = 5, F = 1.0, and ϕ₀ = 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 = 0.5/ε. The dashed curves represent the topographic trough (the ocean), and the solid curves represent the topographic ridges (the continents). Since we intend to discuss the interaction between the storm-track eddies and blocking over the oceans, an incipient block (an initial soliton) is assumed to be located in the central part of the ocean (downstream of short waves) (Holopainen and Fortelius 1987; Cash and Lee 2000). This helps us clarify the relationship between storm-track eddies and downstream blocking over the oceans.

Let us now consider the initial condition of soliton form

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where A₀ is the value of A(Z, 0) at Z = 0. Note that the value of A₀ 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 A₀ = 0.35/ε, h′₀ = 0.5/ε, μ = 2.4, ε = 0.24, and X₀ = 2.87ε/2, then the time evolution of M(T) is shown in Fig. 3 for a′₀ = 0.17/ε^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 = 0 and h′₀ = 0.5/ε, the planetary-scale field (ψ_e = −u₀y + 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, ∇²ψ̃′₀)_P induced by the prespecified transient eddies (ψ̃′₀) as described in Luo et al. (2001), who found that the synoptic-scale eddies can play either a positive role in the onset stage of blocking or a negative role in the decay stage, which is supported by the numerical experiments carried out by Anderson (1995). A solitary block similar to Fig. 4a is also excited through the eddy forcing from the preexisting eddies upstream, even if the constant amplitude [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 (ψ̃₀) 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 = 0.5/ε) with the synoptic-scale eddies upstream. This case may correspond to a northeast Pacific block event. The incipient blocking ridge is constructed to consist of two parts: a slowly varying dipole component and topographically induced monopole component. Thus, in the block–eddy interaction the dipole component is forced directly by the preexisting eddies upstream but not influenced by the topography. However, the superposition of the two components makes the blocking asymmetric, even becoming an omega-type blocking high, depending on the relative amplitudes and phase of the topographically induced standing ridge and the amplified dipole soliton. It should be pointed out that in the absence of synoptic-scale eddies no blocking circulation can be established through the forcing of large-scale topography in a weak westerly background (not shown). This is inconsistent with the result obtained by Charney and DeVore (1979), and is because the background westerly wind in the North Pacific and Atlantic is usually so weak (Lejenäs and Madden 1992) that the resonance condition proposed by Tung and Lindzen (1979) and the topographic instability condition proposed by Charney and DeVore (1979) are not easily satisfied. Under the long-wave approximation, Warn and Brasnett (1983) found that soliton-like blocking could be amplified and captured by small-amplitude isolated topography. However, as we will see later, the topographically induced standing ridge that is in phase with the dipole wave of incipient block seems to control the pattern and location of the formed blocking high and to determine the paths of storm tracks.

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′, ∇²ψ + h) + J(ψ, ∇²ψ′) 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′, ∇²ψ + h) + J(ψ, ∇²ψ′) = 0, which corresponds to ψ̃′₁ = 0 in (18a)] for the same parameters as in Fig. 5.

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.