• Anderson, J. L., 1995: A simulation of atmospheric blocking with a forced barotropic model. J. Atmos. Sci, 52 , 25932607.

  • Berggren, R., , B. Bolin, , and C. G. Rossby, 1949: An aerological study of zonal motion, its perturbations and break-down. Tellus, 1 , 1437.

    • Search Google Scholar
    • Export Citation
  • Butchart, N., , K. Haines, , and J. C. Marshall, 1989: A theoretical study of solitary waves and atmospheric blocking. J. Atmos. Sci, 46 , 20632078.

    • Search Google Scholar
    • Export Citation
  • Cai, M., , and H. M. Van den Dool, 1991: Low-frequency waves and traveling storm tracks. Part I: Barotropic component. J. Atmos. Sci, 48 , 14201436.

    • Search Google Scholar
    • Export Citation
  • Cash, B. A., , and S. Lee, 2000: Dynamical processes of block evolution. J. Atmos. Sci, 57 , 32023218.

  • Charney, J. G., , and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci, 36 , 12051216.

  • Colucci, S. J., 1985: Explosive cyclogenesis and large-scale circulation changes: Implications for atmospheric blocking. J. Atmos. Sci, 42 , 27012717.

    • Search Google Scholar
    • Export Citation
  • Egger, J., 1978: Dynamics of blocking high. J. Atmos. Sci, 35 , 17881801.

  • 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 , 16912708.

    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., 1982: A unified three-dimensional instability theory of the onset of blocking and cyclogenesis. J. Atmos. Sci, 39 , 969987.

    • Search Google Scholar
    • Export Citation
  • Green, J. S. A., 1977: The weather during July 1977: Some dynamical consideration of the drought. Weather, 32 , 120126.

  • Haines, K., , and J. C. Marshall, 1987: Eddy-forced coherent structures as a prototype of atmospheric blocking. Quart. J. Roy. Meteor. Soc, 113 , 681704.

    • Search Google Scholar
    • Export Citation
  • Haines, K., , and A. J. Holland, 1998: Vacillation cycles and blocking in a channel. Quart. J. Roy. Meteor. Soc, 124 , 873895.

  • 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 , 32383259.

    • Search Google Scholar
    • Export Citation
  • Holopainen, E., , and C. Fortelius, 1987: High-frequency transient eddies and blocking. J. Atmos. Sci, 44 , 16321645.

  • Illari, L., , and J. C. Marshall, 1983: On the interpretation of eddy fluxes during a blocking episode. J. Atmos. Sci, 40 , 22322242.

  • 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 , 590596.

    • Search Google Scholar
    • Export Citation
  • Lejenäs, H., , and H. Økland, 1983: Characteristics of Northern Hemisphere blocking as determined from a long time series of observed data. Tellus, 35A , 350362.

    • Search Google Scholar
    • Export Citation
  • Lejenäs, H., , and R. A. Madden, 1992: Traveling planetary-scale waves and blocking. Mon. Wea. Rev, 120 , 28212830.

  • 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 , 2774.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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 , 3179531816.

    • Search Google Scholar
    • Export Citation
  • 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 , 575596.

    • Search Google Scholar
    • Export Citation
  • Malanotte-Rizzoli, P., , and P. Malguzzi, 1987: Coherent structures in a baroclinic atmosphere. Part III: Block formation and eddy forcing. J. Atmos. Sci, 44 , 24932505.

    • Search Google Scholar
    • Export Citation
  • Malguzzi, P., 1993: An analytical study on the feedback between large- and small-scale eddies. J. Atmos. Sci, 50 , 14291436.

  • McIntyre, M. E., , and T. N. Palmer, 1983: Breaking planetary waves in the stratosphere. Nature, 305 , 593600.

  • Nakamura, H., , and J. M. Wallace, 1993: Synoptic behavior of baroclinic eddies during the blocking onset. Mon. Wea. Rev, 121 , 18921903.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., 1998: Poleward deflection of storm tracks. J. Atmos. Sci, 55 , 25772602.

  • Pedlosky, J., 1981: Resonant topographic waves in barotropic and baroclinic flows. J. Atmos. Sci, 38 , 26262641.

  • Shepherd, T. G., 1987: A spectral view of nonlinear fluxes and stationary–transient interaction in the atmosphere. J. Atmos. Sci, 44 , 11661178.

    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1983: The propagation of eddies in diffluent jetstreams: Eddy vorticity forcing of blocking flow fields. Quart. J. Roy. Meteor. Soc, 109 , 737761.

    • Search Google Scholar
    • Export Citation
  • Stewartson, K., 1978: The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn, 9 , 185200.

  • 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 , 203230.

    • Search Google Scholar
    • Export Citation
  • Tsou, C. S., , and P. J. Smith, 1990: The role of synoptic/planetary-scale interactions during the development of a blocking anticyclone. Tellus, 42A , 174193.

    • Search Google Scholar
    • Export Citation
  • 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 , 714734.

    • Search Google Scholar
    • Export Citation
  • 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 , 28452867.

    • Search Google Scholar
    • Export Citation
  • 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 , 28112843.

    • Search Google Scholar
    • Export Citation
  • Warn, T., , and H. Warn, 1978: The evolution of a nonlinear critical layer. Stud. Appl. Math, 59 , 3771.

  • 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 , 2838.

    • Search Google Scholar
    • Export Citation
  • 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 , 19912006.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Evolution of the low-pass filtered 500-hPa height anomalies (thin contours) and the bandpass filtered 500-hPa heights (thick contours, shaded) for a composite of Pacific blocking anticyclones at 2-day intervals from 4 days before block onset to 10 days after onset. Taken from Higgins and Schubert (1994)

  • View in gallery

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

  • View in gallery

    Time evolution of M(T) = |A(Z, T)|max for the parameters A0 = 0.35/ε, h0 = 0.5/ε, μ = 2.4, a0 = 0.17/ε3/2, X0 = 2.87ε/2, and ε = 0.24

  • View in gallery

    Interaction of an initial envelope Rossby soliton (an incipient block) with preexisting synoptic-scale eddies without topography effect for the parameters A0 = 0.35/ε, a0 = 0.17/ε3/2, μ = 2.4, X0 = 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

  • View in gallery

    (Continued)

  • View in gallery

    As in Fig. 4 but with topography (h0 = 0.5/ε)

  • View in gallery

    (Continued)

  • View in gallery

    Instantaneous fields (ψT) of topographically induced eddies during the interaction between preexisting synoptic-scale eddies and prescribed topography for the same parameters as in Fig. 5; CI = 0.1

  • View in gallery

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

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A Barotropic Envelope Rossby Soliton Model for Block–Eddy Interaction. Part I: Effect of Topography

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  • 1 Laboratory of Physical Oceanography, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, China
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Abstract

A new forced envelope Rossby soliton model in an equivalent barotropic beta-plane channel is proposed to describe the interaction between an incipient block (planetary scale) and short synoptic-scale eddies. This model is based on two assumptions, motivated by observations that (i) there exists a zonal scale separation between the planetary-scale and synoptic-scale waves and (ii) that the range of synoptic-scale zonal wavenumber is comparable to the planetary-scale zonal wavenumber. These assumptions allow an analytical treatment. The evolution of the planetary-scale block under the influence of synoptic-scale eddies is described by a forced nonlinear Schrödinger equation that is solved numerically, while the feedback of block development on the preexisting synoptic-scale eddies is derived analytically. It is shown that the planetary-scale projection of the nonlinear interaction between synoptic-scale eddies is the most important contributor to the amplification and decay of the planetary-scale blocking dipole or anticyclone, while the synoptic–planetary-scale interaction contributes significantly to the downstream development of preexisting synoptic-scale eddies. Large-scale topography plays a secondary role compared to the synoptic-scale eddies in exciting the block. However, it plays a role in inducing a standing planetary-scale ridge prior to block onset, which fixes the geographical location of the block and induces meridional asymmetry in the flow. In particular, the topographically induced planetary-scale ridge that is almost in phase with a dipole component of blocking flow is found to be a controlling factor for the northward deflection of storm tracks associated with blocking anticyclones.

Corresponding author address: Dr. Dehai Luo, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, 266003, China. Email: ldh@ouc.edu.cn

Abstract

A new forced envelope Rossby soliton model in an equivalent barotropic beta-plane channel is proposed to describe the interaction between an incipient block (planetary scale) and short synoptic-scale eddies. This model is based on two assumptions, motivated by observations that (i) there exists a zonal scale separation between the planetary-scale and synoptic-scale waves and (ii) that the range of synoptic-scale zonal wavenumber is comparable to the planetary-scale zonal wavenumber. These assumptions allow an analytical treatment. The evolution of the planetary-scale block under the influence of synoptic-scale eddies is described by a forced nonlinear Schrödinger equation that is solved numerically, while the feedback of block development on the preexisting synoptic-scale eddies is derived analytically. It is shown that the planetary-scale projection of the nonlinear interaction between synoptic-scale eddies is the most important contributor to the amplification and decay of the planetary-scale blocking dipole or anticyclone, while the synoptic–planetary-scale interaction contributes significantly to the downstream development of preexisting synoptic-scale eddies. Large-scale topography plays a secondary role compared to the synoptic-scale eddies in exciting the block. However, it plays a role in inducing a standing planetary-scale ridge prior to block onset, which fixes the geographical location of the block and induces meridional asymmetry in the flow. In particular, the topographically induced planetary-scale ridge that is almost in phase with a dipole component of blocking flow is found to be a controlling factor for the northward deflection of storm tracks associated with blocking anticyclones.

Corresponding author address: Dr. Dehai Luo, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, 266003, China. Email: ldh@ouc.edu.cn

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

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)
i1520-0469-62-1-5-e1
where ΨT is the streamfunction, F = (L/Rd)2, and β = β0L2/U; here Rd is the Rossby deformation radius, β0 is the meridional gradient of the Coriolis parameter;
i1520-0469-62-1-5-eq1
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 ΨT into three parts: ΨT = 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 n(n = 1, 2) corresponding to the frequencies ω̃n respectively. We now assume (i) that kn and (ii) that the range of n, Δk̃, satisfies Δkn. 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 n ± k and frequencies ω̃n ± ω. For quasi-stationary planetary-scale (zonal wavenumbers 1–2) wave ψ satisfying |ω| ≪ |ω̃n|, we can have ω̃n ± ωω̃n. 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 n is negligible because it has zonal wavenumber 1 + 2(1 + 2n). 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):
i1520-0469-62-1-5-e2a
The lateral boundary conditions at the two rigid walls are (Pedlosky 1981)
i1520-0469-62-1-5-e3
where ψ = limX→∞ (1/2X) XX ψ dx is the zonally averaged part of the streamfunction ψ and Ly 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′ = ε3/2ψ̃′ 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 = ε5/2ψS 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
i1520-0469-62-1-5-e4a
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:
i1520-0469-62-1-5-e5
We expand the fields ψ̃ and ψ̃′ as
i1520-0469-62-1-5-e6a

Here we consider the special case that the amplitudes of the preexisting synoptic-scale eddies ψ̃0 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, ψ̃1 inevitably contains slow time and spatial variables T1, T2, X1, and X2 due to the feedback of planetary-scale block flow ψ̃0, even if ψ̃0 can be assumed to be independent of slow variables.

Using (5) and (6a) and (6b), (4b) yields the first two orders in ε:
i1520-0469-62-1-5-e7a
It is clear that to first-order approximation, the preexisting synoptic-scale eddies (ψ̃0) are linear free Rossby waves, but the second-order solution (ψ̃1) in (7b) is induced by the feedback of block development and is a modification to the preexisting synoptic-scale eddies (ψ̃0) if
i1520-0469-62-1-5-eq2
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 (ψ̃0) 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)
i1520-0469-62-1-5-e8a
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, 1 and 2, whose time scales are shorter than one week in order to assure that the eddy forcing J(ψ̃0, ∇2ψ̃0)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 (ψ̃0) and synoptic-scale vorticity source (ψ̃S) prescribed in this paper are thus assumed to be
i1520-0469-62-1-5-e9a
where m = −2π/Ly, ω̃1 = u01 − (β + Fu0)1/21, ω̃2 = u02 − (β + Fu0)2/22, 2n = 2n + m2/4 + F, 1 = (ñ − Δñ)k0, 2 = (ñ + Δñ)k0, Δññ, 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 B1 and B2 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 B1 = B2 = f ′0 = a0 exp[−μ(X1 + X0)2] if f ′1 and f ′2 are prescribed by
i1520-0469-62-1-5-eq3
from the nonsecular term equation derived in section 2b, where a0 is the amplitude of the preexisting synoptic-scale eddies, X0 = εxo represents the position of the maximum amplitude of preexisting eddies upstream, and μ denotes the distribution of these eddies.
In the Northern Hemisphere (NH) mid–high latitudes, the large-scale land–sea contrast topography exhibits approximately a wavy topography having zonal wavenumber 2 and monopole meridional structure (Charney and DeVore 1979). In this paper, an incipient block is assumed to be located in the topographic trough (ocean) in order to emphasize the instantaneous role of synoptic-scale eddies in the life cycles of blocking anticyclones occurring over the oceans (Nakamura and Wallace 1993). The bottom topography may be chosen to be of the following form (Pedlosky 1981)
hh0ikxmy
where h0 is the constant amplitude of the wavy topography, k = 2k0 is the zonal wavenumber of the wavenumber two topography, and m = −2π/Ly.

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:
i1520-0469-62-1-5-e11
where
i1520-0469-62-1-5-eq4
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 u0β/(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
i1520-0469-62-1-5-e12a
where
i1520-0469-62-1-5-eq5
Obviously, a special solution to (12b) is
ψ̃1ψ2y,T1T2X1X2
A zonally averaging of (8c) yields an equation for ψ2 and its solution can be obtained as
i1520-0469-62-1-5-e14
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, Ly = 5 (5000 km in dimensional form), F = 1, u0 = β/(k2 + m2), ñ = 10 and Δñ = 0.75, both |k − (21)| ≪ 1 and ω̃2ω̃1ω ≪ 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 − (21) = εΔK and ω̃2ω̃1ω = εΩ(ω = 0 in this paper) and inserting (9), (10), (11), (12), and (14) into (8c), a forced NLS equation can be obtained as
i1520-0469-62-1-5-e15
where the coefficients are given in the appendix.
If one introduces the coordinates
ZX1CgT1TT1
then Eq. (15) can be rewritten as
i1520-0469-62-1-5-e17

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

Substitution of (9a), (10), and (11) into (7b) yields
i1520-0469-62-1-5-e18a
where the coefficients in (18) are given in the appendix.

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 (ψB) and topography effect (ψT). In other words, the dipole component of a developing block seems to dominate the change of modulated synoptic-scale eddies because the topography with a monopole meridional structure is fixed, but the large-scale topography seem to affect the meridional structure of synoptic-scale eddies. It must be pointed out that because ψB contains the amplitude A of the dipole wave and the synoptic-scale eddy wavenumber and amplitude, ψB becomes larger so that εψ̃1 is the same as or exceeds ψ̃0 when the block flow grows and is within a strong stage. In this case, an asymptotic theory is violated because the strongly nonlinear process is dominant. This is a difficult problem encountered in a weakly nonlinear theory. However, we can choose the parameters such as the initial amplitude of the block, the zonal wavenumber, and the intensity of preexisting eddies to make εψ̃1 smaller than or have the same order as ψ̃0 (not shown). Even so, the results, as obtained in the next section, are able to explain the behavior of observed synoptic-scale wave activity during the blocking episode.

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 ΔT = 0.0005 and ΔZ = 0.08, respectively, over the space domain of −20 ≤ Z ≤ 20. The total time of the model run is T = 2 × 104ΔT = 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 f0 = 10−4 s−1, D = 104 m, L = 106 m, U = 10 m s−1, and h = 1000 m, ε = 0.24, Ly = 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 h0 = 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
i1520-0469-62-1-5-e19
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, h0 = 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 a0 = 0.17/ε3/2.

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 h0 = 0 and h0 = 0.5/ε, the planetary-scale field (ψe = −u0y + ψ), synoptic-scale field (ψ′), and the total field (ΨT = ψe + 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 JP = −J(ψ̃0, ∇2ψ̃0)P induced by the prespecified transient eddies (ψ̃0) 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 (JP), 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 h0 = 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 ψT.

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 ψ̃1 = 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.

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:

  1. 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.
  2. 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.
  3. 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 , 25932607.

  • Berggren, R., , B. Bolin, , and C. G. Rossby, 1949: An aerological study of zonal motion, its perturbations and break-down. Tellus, 1 , 1437.

    • Search Google Scholar
    • Export Citation
  • Butchart, N., , K. Haines, , and J. C. Marshall, 1989: A theoretical study of solitary waves and atmospheric blocking. J. Atmos. Sci, 46 , 20632078.

    • Search Google Scholar
    • Export Citation
  • Cai, M., , and H. M. Van den Dool, 1991: Low-frequency waves and traveling storm tracks. Part I: Barotropic component. J. Atmos. Sci, 48 , 14201436.

    • Search Google Scholar
    • Export Citation
  • Cash, B. A., , and S. Lee, 2000: Dynamical processes of block evolution. J. Atmos. Sci, 57 , 32023218.

  • Charney, J. G., , and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci, 36 , 12051216.

  • Colucci, S. J., 1985: Explosive cyclogenesis and large-scale circulation changes: Implications for atmospheric blocking. J. Atmos. Sci, 42 , 27012717.

    • Search Google Scholar
    • Export Citation
  • Egger, J., 1978: Dynamics of blocking high. J. Atmos. Sci, 35 , 17881801.

  • 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 , 16912708.

    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., 1982: A unified three-dimensional instability theory of the onset of blocking and cyclogenesis. J. Atmos. Sci, 39 , 969987.

    • Search Google Scholar
    • Export Citation
  • Green, J. S. A., 1977: The weather during July 1977: Some dynamical consideration of the drought. Weather, 32 , 120126.

  • Haines, K., , and J. C. Marshall, 1987: Eddy-forced coherent structures as a prototype of atmospheric blocking. Quart. J. Roy. Meteor. Soc, 113 , 681704.

    • Search Google Scholar
    • Export Citation
  • Haines, K., , and A. J. Holland, 1998: Vacillation cycles and blocking in a channel. Quart. J. Roy. Meteor. Soc, 124 , 873895.

  • 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 , 32383259.

    • Search Google Scholar
    • Export Citation
  • Holopainen, E., , and C. Fortelius, 1987: High-frequency transient eddies and blocking. J. Atmos. Sci, 44 , 16321645.

  • Illari, L., , and J. C. Marshall, 1983: On the interpretation of eddy fluxes during a blocking episode. J. Atmos. Sci, 40 , 22322242.

  • 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 , 590596.

    • Search Google Scholar
    • Export Citation
  • Lejenäs, H., , and H. Økland, 1983: Characteristics of Northern Hemisphere blocking as determined from a long time series of observed data. Tellus, 35A , 350362.

    • Search Google Scholar
    • Export Citation
  • Lejenäs, H., , and R. A. Madden, 1992: Traveling planetary-scale waves and blocking. Mon. Wea. Rev, 120 , 28212830.

  • 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 , 2774.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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 , 3179531816.

    • Search Google Scholar
    • Export Citation
  • 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 , 575596.

    • Search Google Scholar
    • Export Citation
  • Malanotte-Rizzoli, P., , and P. Malguzzi, 1987: Coherent structures in a baroclinic atmosphere. Part III: Block formation and eddy forcing. J. Atmos. Sci, 44 , 24932505.

    • Search Google Scholar
    • Export Citation
  • Malguzzi, P., 1993: An analytical study on the feedback between large- and small-scale eddies. J. Atmos. Sci, 50 , 14291436.

  • McIntyre, M. E., , and T. N. Palmer, 1983: Breaking planetary waves in the stratosphere. Nature, 305 , 593600.

  • Nakamura, H., , and J. M. Wallace, 1993: Synoptic behavior of baroclinic eddies during the blocking onset. Mon. Wea. Rev, 121 , 18921903.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., 1998: Poleward deflection of storm tracks. J. Atmos. Sci, 55 , 25772602.

  • Pedlosky, J., 1981: Resonant topographic waves in barotropic and baroclinic flows. J. Atmos. Sci, 38 , 26262641.

  • Shepherd, T. G., 1987: A spectral view of nonlinear fluxes and stationary–transient interaction in the atmosphere. J. Atmos. Sci, 44 , 11661178.

    • Search Google Scholar
    • Export Citation
  • Shutts, G. J., 1983: The propagation of eddies in diffluent jetstreams: Eddy vorticity forcing of blocking flow fields. Quart. J. Roy. Meteor. Soc, 109 , 737761.

    • Search Google Scholar
    • Export Citation
  • Stewartson, K., 1978: The evolution of the critical layer of a Rossby wave. Geophys. Astrophys. Fluid Dyn, 9 , 185200.

  • 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 , 203230.

    • Search Google Scholar
    • Export Citation
  • Tsou, C. S., , and P. J. Smith, 1990: The role of synoptic/planetary-scale interactions during the development of a blocking anticyclone. Tellus, 42A , 174193.

    • Search Google Scholar
    • Export Citation
  • 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 , 714734.

    • Search Google Scholar
    • Export Citation
  • 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 , 28452867.

    • Search Google Scholar
    • Export Citation
  • 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 , 28112843.

    • Search Google Scholar
    • Export Citation
  • Warn, T., , and H. Warn, 1978: The evolution of a nonlinear critical layer. Stud. Appl. Math, 59 , 3771.

  • 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 , 2838.

    • Search Google Scholar
    • Export Citation
  • 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 , 19912006.

    • Search Google Scholar
    • Export Citation

APPENDIX

Coefficients for the NLS Equations and Deformed Eddies

The coefficients in (14) are defined by
i1520-0469-62-1-5-eqa1
The coefficients in (15) are defined by
i1520-0469-62-1-5-eqa2
The coefficients in (18) are defined by
i1520-0469-62-1-5-eqa3

Fig. 1.
Fig. 1.

Evolution of the low-pass filtered 500-hPa height anomalies (thin contours) and the bandpass filtered 500-hPa heights (thick contours, shaded) for a composite of Pacific blocking anticyclones at 2-day intervals from 4 days before block onset to 10 days after onset. Taken from Higgins and Schubert (1994)

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

Fig. 2.
Fig. 2.

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

Fig. 3.
Fig. 3.

Time evolution of M(T) = |A(Z, T)|max for the parameters A0 = 0.35/ε, h0 = 0.5/ε, μ = 2.4, a0 = 0.17/ε3/2, X0 = 2.87ε/2, and ε = 0.24

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

Fig. 4.
Fig. 4.

Interaction of an initial envelope Rossby soliton (an incipient block) with preexisting synoptic-scale eddies without topography effect for the parameters A0 = 0.35/ε, a0 = 0.17/ε3/2, μ = 2.4, X0 = 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

Fig. 4.
Fig. 4.

(Continued)

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

Fig. 5.
Fig. 5.

As in Fig. 4 but with topography (h0 = 0.5/ε)

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

Fig. 5.
Fig. 5.

(Continued)

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

Fig. 6.
Fig. 6.

Instantaneous fields (ψT) of topographically induced eddies during the interaction between preexisting synoptic-scale eddies and prescribed topography for the same parameters as in Fig. 5; CI = 0.1

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

Fig. 7.
Fig. 7.

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

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