A Perspective on the Evolution of Atmospheric Blocking Theories: From Eddy-Mean flow Interaction to Nonlinear Multiscale Interaction

Atmospheric blocking is an important quasi-stationary low-frequency dipole mode with a lifetime of 10–20 days, mainly occurring in mid-high latitudes (Berggren et al., 1949; Rex, 1950), which leads to stratospheric warming (Quiroz, 1986; Martius et al., 2009), summer heat waves (Li et al., 2020; Wang and Luo, 2020), winter cold extremes (Chen and Luo, 2017; Yao et al., 2017; Kautz et al., 2022), Arctic warming (Luo et al., 2016), and Greenland ice sheet melting (Wang and Luo, 2022). Thus, the characteristics and underlying mechanisms of atmospheric blocking have been a research topic (Yeh, 1949; Rex, 1950; Sumner, 1959; Kikuchi, 1971; White and Clark, 1975; Egger, 1978; Charney and Devore, 1979; Tung and Lindzen, 1979; Hartmann and Ghan, 1980; Kalnay-Rivas and Merkine, 1981; Frederiksen, 1982; Hoskins et al., 1983; Shutts, 1983; Colucci, 1985, 1987; Haines and Marshall, 1987; Holopainen and Fortelius, 1987; Tsou and Smith, 1990; Lupo and Smith, 1995; Nakamura et al., 1997; Luo, 2000, 2005; Diao et al., 2006; Luo et al., 2014, 2019a; Nakamura and Huang, 2018; Woollings et al., 2018; Zhang and Luo, 2020; Luo and Zhang, 2020a, b, 2021; Lupo, 2021; Kautz et al., 2022).

It has been recognized that the formation and maintenance of atmospheric blocking are not only related to topographic forcing and land-sea contrast (Kikuchi, 1971; Egger, 1978; Charney and Devore, 1979; Tung and Lindzen, 1979; Kalnay-Rivas and Merkine, 1981), but also related to the forcing of transient synoptic-scale eddies (Berggren et al., 1949; Ji and Tibaldi, 1983; Shutts, 1983; Colucci, 1985, 1987; Haines and Marshall, 1987; Holopainen and Fortelius, 1987; Tsou and Smith, 1990; Lupo and Smith, 1995; Nakamura et al., 1997; Mu and Jiang, 2008). The numerical experiments of Ji and Tibaldi (1983) indicated that the forcing of synoptic-scale eddies might play a more important role in the blocking formation than the topographic forcing. Shutts (1983) proposed an eddy straining mechanism to suggest that the eddy straining around the two sides of the blocking region is important for the maintenance of blocking. Haines and Marshall (1987) also found that the time-mean eddy vorticity flux tends to sustain the blocking dipole against the dissipation. However, Arai and Mukougawa (2002) had doubts about the effectiveness of the eddy straining mechanism in the blocking maintenance.

In past two decades, Luo and his collaborators proposed and developed a nonlinear multi-scale interaction (NMI) model to examine the physical mechanism of the life cycle (onset, growth and decay) of atmospheric blocking (Luo, 2000, 2005; Luo et al., 2014, 2019a; Luo and Zhang, 2020a, b, 2021; Zhang and Luo, 2020). The greatest advantage of this NMI model is that it can efficiently describe the lifecycle of atmospheric blocking from the onset and growth to the disappearance compared to previous theories. Based on this NMI model, it is revealed that the eddy deformation (e.g., eddy straining, cyclonic wave breaking, and eddy merging) is not important for the blocking formation and its maintenance. Instead, it is a result of the feedback of the blocking formation itself. The onset, growth, and decay of blocking is mainly the result of the spatiotemporal evolution of pre-existing upstream synoptic-scale eddies as they interact with the initial blocking (Luo, 2000, 2005; Mu and Jiang, 2008; Jiang and Wang, 2010; Ma and Liang, 2017). In this paper, we comprehensively review the previous theoretical studies and present a recent perspective on the dynamical mechanisms of atmospheric blocking. On this basis, we emphasize that the eddy straining mechanism of Shutts (1983), as a milestone work of the blocking theory, is probably incorrect for the formation and maintenance of atmospheric blocking because the time-mean eddy-mean flow interaction model used in his paper cannot disclose the causal relationship between the blocking formation and the eddy deformation. Moreover, some observational evidence is also provided to support our new viewpoint obtained from the NMI model.

The remainder of this paper is organized as follows. In section 2, we describe the data and methodology. We present the schematic diagram of a highly idealized blocking event, a North Pacific blocking case, and the composite result of North Pacific blocking events in section 3. The time-mean eddy-mean flow interaction model of atmospheric blocking, used widely in previous studies, is described in section 4. Section 5 presents the barotropic NMI model and its theoretical results. On this basis, the connection of atmospheric blocking and associated weather extremes to climate change is established by using the NMI model in section 6. The conclusion and discussions are summarized in the final section.

This paper utilizes the the European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis Version 5 (EAR5) data with a 1° × 1° horizontal resolution from December 1950–February 1951 to December 2021–February 2022 (hereafter: 1950–2022) (Hersbach et al., 2020; https://climate.copernicus.eu/climate-reanalysis), which includes daily 500-hPa geopotential height (Z500). For a composite, the long-term (1950–2021) mean and linear trend of the daily data for each calendar day are removed to generate de-seasonalized data.

To identify blocking events over the North Pacific (130°E–170°W), we use the one-dimensional blocking index proposed by Tibaldi and Molteni (1990, TM). The TM index is based on the reversal of the meridional gradient of Z500: $ {\text{GHGN}} = [Z500({\phi _{\text{N}}}) - Z500({\phi _0})]/({\phi _{\text{N}}} - {\phi _0}) $ and ${\text{GHGS}} = [Z500({\phi _0}) - Z500({\phi _{\text{S}}})]/({\phi _0} - {\phi _{\text{S}}})$ at three given reference latitudes $ {\phi _{\text{N}}} = {80^ \circ }{\text{N}} + \Delta $, $ {\phi _0} = {60^ \circ }{\text{N}} + \Delta $, $ {\phi _{\text{S}}} = {40^ \circ }{\text{N}} + \Delta $ and Δ = –5°, 0°, 5°. A blocking event is defined to have occurred if the conditions GHGS > 0 and GHGN = –10 gpm (deg lat.)^–1 are satisfied for at least five consecutive days and at least once choice of Δ in a zonal region covering at least 15° of longitude. For a composite, the time of the North Pacific blocking with a maximum amplitude is defined as its peak day (lag 0).

Berggren et al. (1949) first attributed the establishment of a blocking event to the development of unstable waves in a basic westerly current. However, such a blocking situation is thought to have originated from the forcing of high-frequency transient (synoptic-scale) eddies (Green, 1977). Figure 1 shows an idealized sketch of a meandering blocking flow plotted by Berggren et al. (1949).

Figure 1.  Idealized sketches of the development of unstable waves at the 500 hPa level, in association with the development of a blocking anticyclone in high latitudes. The blue color represents the cold air and the red color indicates the warm air. Solid lines are streamlines and broken lines the frontal boundaries (Reprinted from Berggren et al., 1949).

It is found that the establishment of a blocking flow is followed by the northward (southward) shift of intensified small-scale ridges (troughs) in the blocking region, which resembles the formation of a strong meandering westerly jet stream. We also see several isolated anticyclonic and cyclonic vortices within the blocking region. This feature reflects the role of synoptic-scale eddies in the generation of blocking. Such blocking structures can be observed over the North Atlantic (Luo et al., 2019a) and Pacific.

Figure 2 shows the temporal evolution of a North Pacific blocking event from 1–15 January, 2022.

Figure 2.  Time sequences of daily 500-hPa geopotential height (Z500) fields for a North Pacific blocking event occurring during the period from 1 Jan. 2022 as denoted by 20220101 to 15 Jan. 2022 as denoted by 20220115, where the thick black line represents the contour line of 5300 gpm.

It is seen from Fig. 2 that there is a weak blocking ridge near 180°W on 1 January 2022. Accompanying the intensification of this weak ridge, a North Pacific block formed during 4–10 January. The northward (southward) shift of intensified (deepened) synoptic-scale ridges (troughs) is also seen during the growth process of blocking, which reflects a process of cyclonic wave breaking (CWB). We also find that cyclone merging occurs from 3–6 January, which was suggested to be the cause of the atmospheric blocking (Yamazaki and Itoh, 2013). Moreover, the formation of this North Pacific blocking event also characterizes a strong meandering of westerly jet streams over North Pacific, in agreement with the observational picture of the blocking formation (Berggren et al., 1949). Such a blocking structure is also referred to as the Berggren-Bolin-Rossby (BBR) type blocking, which cannot be explained by the models of Charney and Devore (1979), McWilliams (1980) and Shutts (1983). Using the viewpoint of the eddy straining or CWB or eddy merging, it is difficult to establish a theoretical model to describe the lifecycle of the BBR-type blocking because the eddy straining or CWB or eddy merging is, to some extent, a concomitant phenomenon of the blocking occurrence rather than a cause of the blocking formation.

Using the TM blocking index, it is found that there were 161 North Pacific blocking events during 1950–2021. The composite daily Z500 anomaly field of the North Pacific blocking events is shown in Fig. 3. Clearly, this figure reflects the temporal evolution of the planetary-scale anomaly field of the North Pacific blocking.

Figure 3.  2-days interval instantaneous fields of composite daily 500-hPa geopotential height (Z500, contours with the contour interval of 20 gpm) anomalies from lag −8 to 6 days for 161 North Pacific blocking events in winter during 1950−2020, where lag 0 denotes the peak day of blocking.

It is clearly seen from Fig. 3 that the composite North Pacific blocking moves gradually westward along with its intensification. The westward movement of this North Pacific blocking is slow during the period from lag –8 to –4 days, but relatively rapid during the period from lag –4 to 4 days. Thus, the North Pacific blocking shows a non-uniform westward movement speed. Moreover, we can see that the North Pacific blocking possesses an asymmetric dipole structure with a strong anticyclonic anomaly to the north side of the blocking region and a weak cyclonic anomaly to its south side (Fig. 3). The observed North Atlantic blocking has similar characteristics (Luo et al., 2019a). While Fig. 3 reflects the daily planetary-scale anomaly part of the North Pacific blocking event, Fig. 2 is a result of the instantaneous planetary-scale and synoptic-scale components superimposed on the background flow for a blocking flow over the North Pacific. However, the reason why the North Pacific blocking shows a strong westerly jet meandering (Fig. 2) and the composite daily Z500 anomaly has an asymmetric dipole structure (Fig. 3) cannot be explained by the previous theoretical models (Charney and Devore, 1979; McWilliams, 1980; Shutts, 1983; Haines and Marshall, 1987).

Although some blocking events result from the propagation of planetary wave trains (Nakamura et al., 1997), most blocking events are related to the forcing of transient synoptic-scale eddies (Ji and Tabaldi, 1983; Holopainen and Fortelius, 1987). In previous studies, the time-mean eddy-mean flow interaction model based on time scale decomposition is used to examine how synoptic-scale eddies sustain the blocking flow (Hoskins et al., 1983; Shutts, 1983; Illari and Marshall, 1983). In this model, atmospheric blocking is often considered as a mean flow. Here, we briefly describe this model. In this model, it has been assumed that the mean flow has a long or low-frequency period of T, whereas the synoptic-scale eddies have a short period less than $ \tau $ ($ \tau {\text{ }} < {\text{ }}T $). Then, when the non-dimensional total atmospheric streamfunction $ {\psi _T}(x,y,t) $, scaled by characteristic horizontal velocity $ \tilde U $ (~10 m s^–1) and length $ \tilde L $ (~1000 km), is decomposed into the mean flow $ \bar \psi (x,y,t) $ ($ t{\text{ = }}T{\text{ > }}\tau $) and transient synoptic-scale eddies $ \psi '(x,y,t) $ ($ t \leqslant \tau $) in the form of $ {\psi _T}(x,y,t) = \bar \psi (x,y,t) + \psi '(x,y,t) $ for a quasi-geostrophic barotropic flow, the non-dimensional potential vorticity (PV) equation of the time-mean eddy-mean flow interaction can be obtained by:

where $\overline A = \dfrac{1}{\tau }\displaystyle\int_0^\tau {{{A(}}x,y,t{\text{)d}}} t$ denotes a time average, ${\bf{v'}} = ( - {{\partial \psi '} \mathord{\left/ {\vphantom {{\partial \psi '} {\partial y}}} \right. } {\partial y}},{{\partial \psi '} \mathord{\left/ {\vphantom {{\partial \psi '} {\partial x}}} \right. } {\partial x}})$ is the wind vector of the synoptic-scale streamfunction, $ \bar q = f + {\nabla ^2}\bar \psi - F\bar \psi $ is the PV of the mean flow for $ F = {(\tilde L/{R_d})^2} $ where $ {R_d} $ is the radius of Rossby deformation, $\nabla = {\boldsymbol{i}}( {\partial }/{{\partial x}}) + {\boldsymbol{j}} ({\partial }/{{\partial y}})$ is the horizontal gradient operator, $ q' = {\nabla ^2}\psi ' - F\psi ' $ is the synoptic-scale PV, and $ D $ is the dissipation of the mean flow.

Clearly, it is difficult to use Eq. (1) to examine how synoptic-scale eddies reinforce a blocking flow because $- \nabla \cdot \overline {({\bf{v'}}q')}$ is a time-mean field and cannot be explicitly expressed in terms of the mean flow streamfunction $ \bar \psi (x,y,t) $. Some studies have suggested that $ - \nabla \cdot \overline {({\bf{v'}}q')} $ induced by synoptic-scale eddies $ \psi '(x,y,t) $ can counteract the dissipation ($- \nabla \cdot \overline {({\bf{v'}}q')} + D \approx 0$) to sustain a free blocking mode like a modon solution (Pierrehumbert and Malguzzi, 1984; Haines and Marshall, 1987). Also, $- \nabla \cdot \overline {({\bf{v'}}q')}$ can be simply written as $\nabla \cdot \overline {({\bf{v'}}q')} \approx {{\partial y}}\nabla \cdot {\bf{E}}/ {\partial }y$ and ${\bf{E}} = [\overline {({{v'}^2} - {{u'}^2})} , - \overline {u'v'} ]$ is the $ {\bf{E}} $ vector introduced by Hoskins et al. (1983), whose components represent the Reynolds stresses of the synoptic-scale eddies. Clearly, the meridional derivative of the horizontal divergence of the $ {\bf{E}} $ vector can reflect the eddy-induced change of the mean flow. By calculating the $ {\bf{E}} $ vector and the meridional derivative of its divergence, one can assess the contribution of high-frequency eddies to the maintenance of atmospheric blocking. Even so, it is difficult to examine how the upstream synoptic-scale eddies are deformed by intensified blocking from Eq. (1). Also, this model cannot describe the life cycle (movement, intensity change, and period) of eddy-driven blocking and cannot tell us how the blocking is changed under the forcing of upstream synoptic-scale eddies (Colucci, 1985, 1987) because $ \bar \psi (x,y,t) $ is a time-mean field.

Shutts (1983) proposed an eddy straining mechanism (ESM) to explain how a quasi-stationary blocking is maintained by synoptic-scale eddies based on a time-mean eddy-mean flow interaction model. The basic idea of this ESM is that when the eastward travelling synoptic-scale eddies approach a diffluent straining field represented by a stationary blocking dipole, these eddies suffer an east-west compression and a north-south extension and are subsequently split into two branches around the blocking region (Fig. 4). Such an eddy deformation tends to maintain the blocking dipole through producing the dipole pattern of eddy forcing $- \nabla \cdot \overline {({\bf{v'}}q')}$ with the same polarity as the blocking vortex pair. However, Arai and Mukougawa (2002) used numerical experiments to find that the ESM is inefficient in maintaining atmospheric blocking with large meridional scales, as is often observed. In the ESM of Shutts (1983), a key issue we should address is how a given stationary blocking dipole is generated before an eddy straining takes place. To some extent, the ESM is not appropriate for examining how a blocking dipole is established and how it undergoes a life cycle of 10–20 days from the onset and growth to the decay.

Figure 4.  Schematic picture of the meridional straining of eddies propagating into a stationary blocking as an eddy straining mechanism of blocking maintenance (Shutts, 1983).

Although the blocking flow $ \bar \psi (x,y,t) $ (t = T > τ) can be driven by upstream synoptic-scale eddies $ \psi '(x,y,t) $ ($ t \leqslant \tau $), the feedback of intensified blocking can cause the deformation of the synoptic-scale eddies in zonal and meridional directions. As a result, the deformed synoptic-scale eddies possess not only the synoptic-scale period of $ t \leqslant \tau $, but also the low-frequency period of $ T $, the same as that of the blocking dipole. Thus, the coupling between the blocking and synoptic-scale eddies can lead to a result in which the timescale of synoptic-scale eddies cannot be strictly separated from that of the blocking flow during their interaction. In this case, the time-scale decomposition like ${\psi _T}(x,y,t) = \bar \psi (x,y,t) (t = T > \tau ) + \psi '(x,y,t)(t \leqslant \tau )$ is invalid for examining the interaction between the blocking dipole and synoptic-scale eddies. Consequently, one cannot use the time-mean eddy-mean flow interaction model to examine the mutual interaction between a blocking dipole and synoptic-scale eddies nor to assess how a blocking dipole is generated by the eddy forcing from upstream synoptic-scale eddies. The eddy-mean flow interaction model also cannot explain the generation of observed meandering blocking flows (Figs. 1−2). This is the main reason why we developed the NMI model to illustrate the basic physics of eddy-driven blocking during its lifecycle.

This section briefly describes the barotropic NMI model proposed and developed by Luo and his collaborators (Luo, 2000, 2005; Luo et al., 2014, 2019a; Zhang and Luo, 2020). This NMI model was established based on the space scale decomposition. For a given climatological zonal flow $ \bar \psi (y) $, the non-dimensional total streamfunction $ {\psi _T}(x,y,t) $ is decomposed into three spatial scales: background flow $ \bar \psi (y) $ with a zero wavenumber, a planetary-scale blocking dipole anomaly $ \psi (x,y,t) $ with zonal and meridional wavenumbers ($ k $, $ m $), and a synoptic-scale part $ \psi '(x,y,t) $ with zonal and meridional wavenumbers ($ {\tilde k_i} $, $ m $/2) in the form of $ {\psi _T}(x,y,t) = \bar \psi (y) + \psi (x,y,t) + \psi '(x,y,t) $. In this scale composition, the blocking anomaly $ \psi (x,y,t) $ has a small zonal wavenumber, whereas the synoptic-scale eddies have high zonal wavenumbers. Under the zonal scale separation assumption $ {\tilde k_i} \gg k $ (Luo, 2000, 2005; Luo et al., 2019a), the non-dimensional PV equations of planetary- and synoptic-scale disturbances during their interaction in a background zonal flow $ U = - {{\partial \bar \psi (y)} \mathord{\left/ {\vphantom {{\partial \bar \psi (y)} {\partial y}}} \right. } {\partial y}} $ on a β-plane channel with a non-dimensional width of $ {L_y} $ can be expressed as:

The lateral boundaries used in Eq. (2a) are

where $ \left[ {\;} \right] $ denotes a zonal average of $ \psi (x,y,t) $. In Eq. (2), $ {\text{P}}{{\text{V}}_y} = \beta - {U_{yy}} + FU $ denotes the meridional gradient of the background PV, $ F $ is the same as in Eq. (1) and $ {\nabla ^2}\psi _S^ * $ is the synoptic-scale wavemaker that is designed to produce pre-existing synoptic-scale eddies upstream of the incipient blocking. Note that the “$ P $” in $- \nabla \cdot {({\bf{v'}}q')_P}$ means that $- \nabla \cdot ({\bf{v'}}q')$ is taken to have the same spatial structure as that of $ \psi (x,y,t) $.

Equation (2) can describe the different scale coupling between the blocking and synoptic-scale eddies modulated by the background flow, but Eq. (1) cannot. Thus, the NMI model avoids the limitations of the previous time-mean eddy-mean flow interaction model in representing the mutual interaction between the blocking and synoptic-scale eddies. Because the blocking anomaly $ \psi (x,y,t) $ has a low zonal wavenumber, $ k $, it corresponds to a long period, $ T $. When upstream synoptic-scale eddies $ \psi '(x,y,t) $ drive the evolution of the blocking anomaly $ \psi (x,y,t) $ originating from an initial blocking through the forcing of $- \nabla \cdot {({\bf{v'}}q')_P}$ in Eq. (2a), the feedback of intensified blocking can cause a low-frequency change in synoptic-scale eddies approaching the downstream blocking, as seen from Eq. (2b). The presence of the intensified blocking can also induce a splitting of westerly jet streams around the blocking region to generate a low-frequency variation of the mean flow having the same timescale as that of the blocking dipole, even though the background flow $ \bar \psi (y) $ influences the blocking evolution. Thus, the three scale components of the blocking field are coupled together and have the same low-frequency timescale during their interaction process, as shown in Fig. 5 on the schematic diagram.

Figure 5.  Conceptual framework of the interaction of the upstream synoptic-scale eddies with the planetary-scale blocking dipole component $ {\psi _B} $ and background flow leading to the formation of a blocking dipole.

Defining $ q = {\nabla ^2}\psi - F\psi $, one can rewrite Eq. (2a) as

Generally speaking, $ q + {\text{PV}} $ exhibits a linear relationship with $ \psi + \bar \psi $ (Shutts, 1983; Illari and Marshall, 1983) so that $ J(\psi + \bar \psi ,q + {\text{PV)}} \approx {\text{0}} $. In this case, we have ${{\partial q}}/{{\partial t}} \approx $ $- \nabla \cdot {({\bf{v'}}q')_P}$. When $- \nabla \cdot {({\bf{v'}}q')_P}$ matches the spatial structure of the initial blocking $ q $, an initial blocking can be amplified into a typical blocking. This is the so-called eddy-blocking matching (EBM) mechanism of the blocking formation (Luo et al., 2014, 2019a).

The analytical solution to Eq. (2) can be further obtained by assuming $ \psi ' = {\psi '_1} + {\psi '_2} $, where $ {\psi '_1} $ denotes the pre-existing synoptic-scale eddy streamfunction and $ {\psi '_2} $ represents the deformed eddies due to the feedback of intensified blocking. During the initial stage ($ t \sim 0 $) of blocking, one can have ${{\partial q}}/{{\partial t}} \approx - \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ because of $ {\psi '_2} \approx 0 $, where ${\bf{v}}'_1 = ( - {{\partial {\psi '_1}} \mathord{\left/ {\vphantom {{\partial {\psi '_1}} {\partial y}}} \right. } {\partial y}},{{\partial {\psi '_1}} \mathord{\left/ {\vphantom {{\partial {\psi '_1}} {\partial x}}} \right. } {\partial x}})$ and $ {q'_1} = {\nabla ^2}{\psi '_1} - F{\psi '_1} $. Clearly, whether an initial blocking can evolve into a typical blocking depends on the spatial structure of pre-existing synoptic-scale eddies, $ {\psi '_1} $, upstream of the initial blocking. Only when $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ has nearly the same spatial structure as the PV anomaly $ {\left. q \right|_{t = 0}} $ of the initial blocking can a blocking dipole be formed under the forcing of pre-existing synoptic-scale eddies. Clearly, our EBM mechanism is different from the ESM of Shutts (1983), who emphasized the role of eddy straining in the blocking maintenance. In our NMI model, the spatiotemporal evolution of $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ plays a key role in the onset, growth, and decay of the blocking dipole. Because $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ has a low-frequency period of 10–20 days, the formed blocking can inevitably show a lifetime of 10–20 days. Thus, the NMI model can efficiently capture the lifecycle of eddy-driven blocking from the onset and growth to the decay unlike the previous theoretical models (McWilliams, 1980; Shutts, 1983; Haines and Marshall, 1987).

In general, the synoptic-scale eddies in mid-latitudes consist of many zonal modes. In this paper, we assume that the pre-existing synoptic-scale eddies are composed of two waves with wavenumbers $ ({\tilde k_1},m/2) $ and $ ({\tilde k_2},m/2) $, whereas the blocking anomaly has a carrier wave with wavenumbers$ (k,m) $. Here, the blocking anomaly has a meridional dipole structure, whereas the pre-existing synoptic-scale eddies have a monopole structure in the meridional direction. We further suppose that $ U $ and $ {\text{P}}{{\text{V}}_y} $ are slowly varying in the zonal direction so that $ U $ and $ {\text{P}}{{\text{V}}_y} $ can be considered as being constants when compared to the $ x $ and $ y $ variations. As done in Luo et al. (2019a), the analytical solution of the interaction between the blocking anomaly and synoptic-scale eddies in a given background flow $ U $ in the form of fast variables $ (x,y,t) $ can be obtained as:

where $ i = \sqrt { - 1} $, $ B $ denotes the envelope complex amplitude of the blocking anomaly with its complex conjugate $ {B^ * } $ and $ {\left| B \right|^2} = B{B^ * } $, $\omega {\text{ }} = {\text{ }}Uk{\text{ }} - {{{\text{P}}{{\text{V}}_{\text{y}}}k}}/({{{k^2} + {m^2} + F}})$ is the frequency of the blocking carrier wave, ${\tilde \omega _j} = {\text{ }}U{\tilde k_j} - {{{\text{P}}{{\text{V}}_{\text{y}}}{{\tilde k}_j}}}/ $ $ ({{\tilde k}_j}^2 + {m^2}/4 + F)$ is the frequency of each mode of the pre-existing synoptic-scale eddies, $ {f_0}(x) = {a_0}\exp [ - \mu {\varepsilon ^2}{(x + {x_T})^2}] $ is the slowly varying distribution of pre-existing synoptic-scale eddies with a constant eddy amplitude of $ {a_0} $ for $ \mu > 0 $ and $ \varepsilon \ll 0 $, where $ {x_T} $ denotes the upstream location of the pre-existing synoptic-scale eddies relative to the initial blocking at $ x = 0 $. Note that $ {\text{cc}} $ denotes the complex conjugate of its preceding term. We also allow $ {\alpha _1} = 1 $, $ {\alpha _2} = \alpha = - 1 $ and $ m = - 2\pi /{L_y} $ in Eq. (3). The mathematical expressions of the coefficients $ {q_n} $, $ {g_n} $, $ {Q_j} $, $ {p_j} $, $ {r_j} $, $ {s_j} $, and $ {h_j} $ can be found in Luo et al. (2019a)

The envelope amplitude $ B(x,t) $ of the eddy-driven blocking anomaly satisfies the following forced nonlinear Schrӧdinger (NLS) equation:

where $ \Delta k = k - ({\tilde k_2} - {\tilde k_1}) $, $ \Delta \omega = {\tilde \omega _2} - {\tilde \omega _1} - \omega $, and ${C_g} = U - {{{\text{P}}{{\text{V}}_y}({m^2} + F - {k^2})}}/{{{{({k^2} + {m^2} + F)}^2}}}$. The dispersion term is given by $ \lambda = {\text{P}}{{\text{V}}_y}{\lambda _0} $ with ${\lambda _0} = {{[3({m^2} + F) - {k^2}]k}}/ {{{{({k^2} + {m^2} + F)}^3}}}$; $ \delta = {\delta _N}{\text{/P}}{{\text{V}}_y} $ is the nonlinearity term and the coefficients $ {\delta _N} $ being positive and $ G $ can be found in Luo et al. (2019a).

Clearly, the blocking anomaly can be represented by a wave packet described by the forced NLS Eq. (4). In Eq. (3) $ {\psi _B} $ denotes the blocking dipole anomaly streamfunction, $ {\psi _m} $ represents the blocking-induced change of the mean zonal flow, $ {\psi '_1} $ is the pre-existing upstream synoptic-scale eddies and $ {\psi '_2} $ is the deformed eddies induced by the blocking evolution. It is noted that the deformed eddies have the same low-frequency timescale as that of blocking because the blocking amplitude $ B(x,t) $ is included in the deformed eddy equation (Eq. 3g). Thus, the pre-existing upstream synoptic-scale eddies tend to have the same low-frequency timescale as that of the blocking anomaly as they interact with the initial blocking.

For a given background flow $ U $, the initial blocking amplitude $ B(x,{\text{ }}0) $ and pre-existing synoptic-scale eddies $ {\psi '_1} $, the spatiotemporal variation of the blocking envelope amplitude $ B(x,{\text{ }}t) $ can be found from the forced NLS equation (Eq. (4)). Thus, one can use Eq. (3) to examine how a blocking dipole is generated by the forcing of upstream synoptic-scale eddies. Naturally, the biases of the background field $ \bar \psi {\text{ }}(y) $, the initial blocking anomaly $ B\;(x,{\text{ }}t) $, and pre-existing synoptic-scale eddies, $ {\psi '_1} $, in numerical models can cause the bias of the blocking prediction in strength, frequency, duration, and location (Tibaldi and Ji, 1983; Nutter et al., 1998). On the other hand, we have $- \Delta \omega = {\text{P}}{{\text{V}}_y} \left[{{\tilde k}_2}/({{\tilde k}_2}^2 + {m^2}/ 4 + F)^2 - {{{{\tilde k}_1}}}/{{{{({{\tilde k}_1}^2 + {m^2}/4 + F)}^2}}} - {k}/({{{{{k^2} + {m^2} + F)}^2}}})\right]$ for $ k = {\tilde k_2} - {\tilde k_1} $. It is clearly found that $ - \Delta \omega $ is small as $ {\text{P}}{{\text{V}}_y} $ is weak. This suggests that the eddy forcing caused by the pre-existing synoptic-scale eddies has a longer duration so that the eddy-driven blocking has a longer lifetime in weaker $ {\text{P}}{{\text{V}}_y} $ regions. In other words, when the upstream synoptic-scale eddies enter the weaker $ {\text{P}}{{\text{V}}_{\text{y}}} $ region, the eddy vorticity forcing becomes more persistent in reinforcing the blocking dipole. We also mention that when $ {\text{P}}{{\text{V}}_y} $ is smaller, the dispersion (nonlinearity) of the blocking system is weaker (stronger) to make the formed blocking more persistent. Thus, a small $ {\text{P}}{{\text{V}}_y} $ is a favorable condition for the maintenance of blocking, even though blocking is driven by upstream synoptic-scale eddies.

Based on Eqs. (3-4), the nonlinear phase speed $ {C_{{\text{NP}}}} $ and energy dispersion speed $ {C_{{\text{ED}}}} $ of the blocking anomaly can be approximately obtained as in Luo et al., (2019a):

where $ {M_0} = \max (\left| B \right|) $ represents the maximum amplitude of the blocking anomaly. In the real application, ${M_0} = \sqrt { {{{L_y}}}/{2}} \max ({\psi _B})/{2}$ is chosen, where $ \max ({\psi _B}) $ represents the maximum amplitude of the blocking anomaly streamfunction. Of course, a domain-averaged value of $ {\psi _B} $ is also considered as the value of $ \max ({\psi _B}) $.

It is clearly seen that the zonal movement speed of the blocking anomaly depends not only on the strength of the background westerly wind ($ U $) and the magnitude of $ {\text{P}}{{\text{V}}_y} $, but also on the blocking amplitude ($ {M_0} $). The blocking anomaly moves more rapidly westward when the blocking amplitude is larger or when $ {\text{P}}{{\text{V}}_y} $ is smaller. Of course, the nonlinear phase speed formula (Eq. 5a) can also be applied to explain the zonal movement of mid-high latitude teleconnection patterns.

The blocking system can have an energy dispersion that is too strong if the blocking amplitude is too large or if the $ {\text{P}}{{\text{V}}_y} $ is too small (Luo et al., 2019a). This implies that blocking with a very large amplitude cannot be maintained. Also, blocking cannot be persistent in a very small ${\text{P}}{{\text{V}}_{y}}$ region. When $ {M_0}\sim 0 $ (e.g., the disturbance amplitude is very small) in a linear framework, one can have a linear phase speed ${C_{{\text{NP}}}} \approx {C_p} = \omega /k = U - {{{\text{P}}{{\text{V}}_y}}}/({{{k^2} + {m^2} + F}})$ and a linear energy dispersion speed ${C_{{\text{ED}}}} \approx {{C}_g} - {C_p} = {{2{k^2} {\text{P}}{{\text{V}}_y}}}/$ $ {{{{({k^2} + {m^2} + F)}^2}}}$. The linear phase speed of the blocking anomaly is clearly related to $ U $ and $ {\text{P}}{{\text{V}}_y} $, whereas the strength of the linear energy dispersion speed is mainly determined by the magnitude of $ {\text{P}}{{\text{V}}_y} $. For a uniform background westerly wind where $ U = {u_0} $, we obtain ${C_{{\text{ED}}}} \approx {{2{k^2}(\beta + {{F}}{u_0})}}/{{{{({k^2} + {m^2} + F)}^2}}}$ (Yeh, 1949). Clearly, there is a weak linear dispersion in the high latitudes or in the weak westerly wind region. This explains why blocking easily occurs in high latitudes. To some extent, the linear dispersive theory of Yeh (1949) is a special case of the NMI model, even though the role of synoptic-scale eddies has been neglected in his theory. We also note that the linear Eliassen-Palm flux cannot be applied to the case of blocking flows because of the strong nonlinear effect of the blocking amplitude.

While $ {\text{P}}{{\text{V}}_{{y}}} $ is an important precursor background factor influencing the evolution of atmospheric blocking, it does not imply that ${\text{P}}{{\text{V}}_{{y}}}$ is fixed during the blocking evolutionary process. In fact, during the lifecycle of blocking, the meridional PV gradient of the planetary scale component of the blocking field is ${({\text{P}}{{\text{V}}_P})_y} = {{\partial ({\text{P}}{{\text{V}}_P})}}/{{\partial {{y}}}} = {\text{P}}{{\text{V}}_y} + {({\text{P}}{{\text{V}}_B})_y} + {({\text{P}}{{\text{V}}_m})_y}$, where ${({\text{P}}{{\text{V}}_B})_y} = {{\partial ({\nabla ^2}{\psi _B} - {{F}}{\psi _B})} \mathord{\left/ {\vphantom {{\partial ({\nabla ^2}{\psi _B} - {{F}}{\psi _B})} {\partial y}}} \right. } {\partial y}}$ is the meridional PV gradient of the blocking anomaly $ ({\psi _B}) $, and ${({\text{P}}{{\text{V}}_m})_y} = {{\partial ({{{\partial ^2}{\psi _m}} \mathord{\left/ {\vphantom {{{\partial ^2}{\psi _m}} {\partial {y^2} - F{\psi _m})}}} \right. } {\partial {y^2} - F{\psi _m})}}} \mathord{\left/ {\vphantom {{\partial ({{{\partial ^2}{\psi _m}} \mathord{\left/ {\vphantom {{{\partial ^2}{\psi _m}} {\partial {y^2} - {\rm{F}}{\psi _m})}}} \right. } {\partial {y^2} - {\rm{F}}{\psi _m})}}} {\partial y}}} \right. } {\partial y}}$ is the meridional PV gradient of the blocking-induced zonally mean zonal wind change. Clearly, the zonally-averaged meridional PV gradient $ {(\overline {{\text{P}}{{\text{V}}_P}} )_y} $ of the blocking field is $ {(\overline {{\text{P}}{{\text{V}}_P}} )_y} = {\text{P}}{{\text{V}}_y} + {\text{ }}{({\text{P}}{{\text{V}}_m})_y} $. Naturally, the zonally averaged meridional PV gradient is altered with blocking evolution. However, there is a condition when $ {(\overline {{\text{P}}{{\text{V}}_P}} )_y} \approx {\text{P}}{{\text{V}}_y} $ happens during the prior period of blocking because $ {({\text{P}}{{\text{V}}_m})_y} \approx 0 $ (Luo et al., 2019a, b). Thus, $ {\text{P}}{{\text{V}}_y} $ can be obtained by calculating the time-mean meridional PV gradient during the prior period of blocking.

The above barotropic NMI model can be extended to the case of a three-dimensional baroclinic atmosphere (Luo and Zhang, 2020a, b, 2021). The NMI model can also be applied to the investigation of the dynamical mechanisms of the North Atlantic Oscillation (NAO) (Luo et al., 2007) and Pacific North American pattern (PNA) (Luo et al., 2020) if $ \alpha = 1 $ and $ m = 2\pi /{L_y} $ are considered. Using this NMI model, one can explore the relationship between the phase of NAO and downstream blocking (Luo et al., 2015a, b, c). It is further revealed that Eurasian blocking (e.g., European blocking and Ural blocking) mainly results from the decay of a positive NAO (NAO⁺) via the propagation of low-frequency wave trains due to intensified energy dispersion from the North Atlantic to Eurasia (Luo et al., 2015c; Yao et al., 2016). Although this paper does not describe the relationship between NAO and downstream blocking, synoptic-scale eddies mostly drive NAO and PNA events. The reason the NAO and PNA patterns show different spatial wave shape structures has not yet been explained in previous studies. However, this issue is easily investigated by the NMI model. In this NMI model, the negative phase NAO (PNA) or NAO^– (PNA^–) event corresponds to a blocking flow over the North Atlantic (Pacific), whereas the positive phase NAO (PNA) or NAO⁺ (PNA⁺) event corresponds to an intensified westerly jet event. Because the background ${\text{P}}{{\text{V}}_{y}}$ is much larger over the North Pacific than over the North Atlantic, the eddy-driven dipole is more easily maintained due to its weak dispersion and strong nonlinearity over the North Atlantic than over the North Pacific. In this case, a localized NAO (wave train-like PNA) structure is easily formed in the North Atlantic (Pacific) (Luo et al., 2020). Thus, blocking, NAO, and PNA events can be unified into the NMI model by choosing different signs for $ \alpha $ and $ m $ as well as different $ {\text{P}}{{\text{V}}_{\text{y}}} $ backgrounds.

Figure 6 shows an idealized schematic diagram of pre-existing upstream synoptic-scale eddies which drive the occurrence of blocking or NAO and PNA events. We find that when $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ features a negative-over-positive dipole structure, the upstream synoptic-scale eddies $ ({\psi '_1}) $ can drive the formation of blocking, NAO^–, and PNA^– events (top of Fig. 6). But when $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ has a positive-over-negative dipole structure, the upstream synoptic-scale eddies $ ({\psi '_1}) $ can lead to the generation of NAO⁺ and PNA⁺ events (bottom of Fig. 6). Of course, the sub-seasonal variability of these teleconnection patterns can be significantly influenced by the magnitude of the background $ {\text{P}}{{\text{V}}_{\text{y}}} $.

Figure 6.  Idealized schematic diagram of pre-existing upstream synoptic-scale eddies ($ {\psi '_1} $) that drive the generation of blocking, NAO, and PNA events through producing $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ in the NMI model, where pre-existing upstream synoptic-scale eddies are referred to as upstream eddies, $ {\psi _B} $ represents the dipole anomaly of blocking, NAO, and PNA, “+” (“−“) denotes the cyclonic (anticyclonic) vorticity forcing and “H” (“L”) indicates high (low) pressure.

In this subsection, as an example, we consider the parameters $ {L_y} = 5 $ (5000 km in the dimensional form), $ k = 2{k_0} $,$ {k_0} = 1/[6.371\cos ({\phi _0})] $, $ U = {u_0} = 0.7 $ (7 m s^–1), $ {a_0} = 0.17 $, $ \mu = 1.2 $, $ {\phi _0} = {55^ \circ }N $, $ {\tilde k_1} = (10 - 0.75){k_0} $, $ {\tilde k_2} = (10 + 0.75){k_0} $, $ \varepsilon = 0.24 $ and $ {x_T} = 2.87/2 $. For an initial amplitude $ B{\text{ }}(x,0) = 0.35 $, the instantaneous streamfunctions of the planetary- (${\psi _P}$) and synoptic-scale ($ \psi ' $) components and their total streamfunction field ($ {\psi _T} = {\psi _P} + \psi ' $) of eddy-driven blocking in a uniform background westerly wind are shown in Fig. 7. It is interesting to see that a weak diffluent flow field appears near $ x = 0 $ at day 0, which represents an initial blocking (Fig. 7a). Because the diffluent flow is weak at day 0, the synoptic-scale eddies at day 0 are not strongly deformed (Fig. 7b). Under the forcing of $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ with a negative-over-positive dipole, this initial blocking can be amplified into a typical blocking dipole with an enhanced retrogression (Fig. 7a; from days 6 to 12).

To some extent, the enhanced retrogression of this blocking is likely related to its increased amplitude in terms of Eq. (5a). When the blocking is strong, the upstream synoptic-scale eddies propagating into the blocking region undergo an east-west compression and north-south extension and are split into two branches around the blocking region (Fig. 7b from days 6 to 12). When the blocking decays, the eddy deformation is weakened and the eddy straining disappears at days 18 and 21 (Fig. 7b). Thus, it is concluded that the eddy straining is likely a result of the feedback of intensified blocking on upstream synoptic-scale eddies. Arai and Mukougawa (2002) also pointed out that the eddy straining is not an effective mechanism of the blocking maintenance. As seen from Figs. 7a-b, the eddy deformation and blocking evolution have the same low-frequency timescale. When the synoptic-scale field (Fig. 7a) is superimposed on the blocking anomaly field (Fig. 7a), the total streamfunction field of the blocking flow shows a strong meandering of westerly jet streams (Fig. 7c), which resembles the life cycle of the idealized BBR-type blocking (Fig. 1) (Luo, 2000, 2005), like a traffic Jam (Nakamura and Huang, 2018) and an observed North Pacific blocking (Fig. 2). At present, only the NMI model can well explain the physical picture of the onset, growth, maintenance, and decay of the BBR-type blocking (Fig. 1). Moreover, the northward (southward) displacement of warm (cold) or low (high) PV air can be seen along with the intensification of the blocking dipole (Fig. 7c; from days 6 to 12), which reflects the occurrence of CWB. But when blocking decays, CWB is weakened and then disappears (Fig. 7c; at days 18 and 21). Thus, the CWB is also a concomitant phenomenon of the blocking formation. Naturally, we cannot conclude that the CWB is a mechanism leading to the formation of blocking. In the NMI model, the blocking is considered as a nonlinear initial value problem (Luo, 2000, 2005), which is confirmed by Mu and Jiang (2008) and Jiang and Wang (2010) using the conditional nonlinear optimal perturbation method.

Because $ U $ and $ {\text{P}}{{\text{V}}_y} $ over North Pacific are weaker in high latitudes than in midlatitudes, the anticyclonic anomaly of the composite blocking dipole over North Pacific (Fig. 3) has inevitably weaker energy dispersion and stronger nonlinearity than the cyclonic anomaly over its south side. As a result, the anticyclonic anomaly of the North Pacific blocking dipole is more easily maintained than its cyclonic anomaly. This is a reason why the anticyclonic anomaly of the North Pacific blocking is stronger than its cyclonic anomaly in the south side of the blocking region. As a result, the retrogression of the anticyclonic anomaly is more rapid than that of the cyclonic anomaly of the North Pacific blocking based on Eq. (5a). In this case, the composite blocking anomaly gradually shows a northwest-southeast tilting during its evolution process. Using a barotropic NMI model, Luo et al. (2019a) presented the theoretical result about how an asymmetric blocking dipole is generated for a basic zonal wind being weaker (stronger) in high latitudes (midlatitudes) (Fig. 8). Clearly, this theoretical result is consistent with the evolution picture (e.g., zonal movement, timescale, and spatial structure) of the observed North Pacific blocking (Fig. 3).

Figure 8.  Instantaneous fields of eddy-driven blocking dipole anomaly streamfunction $ {\psi _B} $ in a basic zonal flow obtained based on a barotropic NMI model for $U = {u_0} + \Delta u{e^{ - \gamma {{(y - {y_0})}^2}}}\cos (\frac{{2\pi }}{{{L_y} + {y_1}}}y)$ with weaker wind speeds in higher latitudes, ${u_0} = 0.7$, $\Delta u = 0.2$ , $\gamma = 0.1$, ${y_0} = 1.5$ and ${y_1} = 3$ with the initial amplitude $B(x,y,0) = 0.4\exp [ - \sigma {\varepsilon ^2}{(y - {y_0})^2}]$ with ${y_0} = 3.75$ and $\sigma = 2$).

In summary, the NMI model can explain not only the life period of observed blocking but also the zonal movement and spatial structure of observed blocking. Moreover, based on the NMI model, we can establish a bridge from climate change to weather extremes. In the next section, we will describe this phenomenon and present a new perspective.

The lifetime, intensity, and movement of atmospheric blocking represent three important factors for whether atmospheric blocking produces strong winter cold extremes and strong summer heat waves. In particular, the lifetime and movement of blocking are the two most key factors. For example, increased quasi-stationarity and persistence of Ural blocking were shown to be important for the generation of strong winter Eurasian cold waves (Yao et al., 2017). Thus, understanding what type of climate change leads to the sub-seasonal variability of atmospheric blocking in lifetime, intensity, and movement has important implications in improving the prediction of weather extremes. It has been recognized that a weather extreme is a rapid process with sub-seasonal timescales (10–20 days) and is closely related to sub-seasonal atmospheric teleconnection pattern events (e.g., blocking, NAO and PNA). However, climate change [e.g., recent Arctic amplification (AA), Atlantic Multi-decadal Oscillation (AMO) and Pacific decadal Oscillation/Interdecadal Pacific Oscillation (PDO/ IPO)] is a slow process compared to sub-seasonal atmospheric teleconnection circulations. Thus, there is a huge gap between weather extremes and climate change. Unfortunately, a theoretical linkage between climate change and weather extremes is not well established in previous studies. However, this issue can be resolved by using the NMI model.

Because the AA, AMO, and PDO/IPO represent slow climate-change phenomena, they mainly change the background condition (e.g., the magnitude of $ {\text{P}}{{\text{V}}_y} $) (Luo et al., 2019a-b, 2022; Chen et al., 2021) rather than the initial blocking and eddy conditions. When $ {\text{P}}{{\text{V}}_y} $ is changed, sub-seasonal atmospheric teleconnection patterns are inevitably altered due to changes in the energy dispersion, nonlinearity, and eddy forcing. In this case, the AA, AMO, and PDO/IPO inevitably modulate weather extremes via the changes in $ {\text{P}}{{\text{V}}_y} $ and associated sub-seasonal teleconnection patterns (Fig. 9). According to Luo et al. (2019a), two different $ {\text{P}}{{\text{V}}_y} $ backgrounds lead to different changes in atmospheric blocking in strength, movement, and duration. A long-lived (short-lived) blocking with large (small) amplitude is easily formed for a small (large) $ {\text{P}}{{\text{V}}_y} $, but the zonal movement of blocking also depends on the background winds and blocking amplitude. For example, AA (Arctic cooling) in Barents-Kara Seas (BKS) corresponds to a small (large) $ {\text{P}}{{\text{V}}_y} $ over the Eurasian mid-high latitudes (Luo et al., 2019b) and a strong (weak) upstream westerly wind (Yao et al., 2017). It is further revealed that a small $ {\text{P}}{{\text{V}}_y} $ favors a long-lived Ural blocking (UB) with no notable zonal movement, whereas a large $ {\text{P}}{{\text{V}}_y} $ promotes a short-lived UB with strong zonal movement (Luo et al., 2019b). Thus, to some extent, the changes in the blocking character under different $ {\text{P}}{{\text{V}}_y} $ backgrounds can explain the impact of climate change on atmospheric blocking through the change in $ {\text{P}}{{\text{V}}_y} $. Clearly, the climate change-blocking linkage could be well established based on the NMI model, even though the statistical connection between climate change and blocking was also discussed in Woollings et al. (2018). However, the different impacts of AMO, PDO/IPO, and their combined effect on $ {\text{P}}{{\text{V}}_y} $ were not examined in detail in previous studies. Consequently, an understanding of the influence of climate change on weather extremes becomes attainable if the nature of the $ {\text{P}}{{\text{V}}_y} $ changes under the climate change conditions are known.

Figure 9.  Schematic diagram of climate change (e.g., AA, AMO, and PDO/IPO) influencing weather extremes due to changes in sub-seasonal teleconnection patterns (e.g., blocking, NAO, PNA, and so on) via the change of the meridional background potential vorticity gradient (PV_y).

In this paper, we first review the previous studies about the dynamical mechanisms of atmospheric blocking based on the eddy-mean flow interaction model and then present a perspective on the onset and formation mechanism of atmospheric blocking from nonlinear multiscale interactions using the NMI model. Thus, this paper reviews the evolution of atmospheric blocking theories from the eddy-mean flow interaction model to the nonlinear multi-scale interaction model and presents some issues about the possible applicability of the NMI model in future work. Previous studies suggested that eddy deformation, such as eddy straining, CWB, and eddy merging, might play important roles in the formation and maintenance of atmospheric blocking (Shutts, 1983; Masato et al., 2013; Yamazaki and Itoh, 2013). These conclusions were both speculative and problematic because the previous theoretical studies based on the time-mean eddy-mean flow interaction model cannot reveal the causal relation between the blocking evolution and eddy deformation (Shutts, 1983; Hoskins et al., 1983). However, the NMI model of Luo (2000, 2005) and Luo et al. (2014, 2019a) can well present the physical picture of the evolution of the BBR-type blocking (Berggren et al., 1949) and avoid the difficulty of the previous theoretical models in representing the lifecycle of eddy-driven blocking, because it considers the formation of atmospheric blocking as a nonlinear initial value problem (Luo, 2000, 2005; Luo et al., 2014, 2019a; Mu and Jiang, 2008). Of course, the NMI model is also appropriate for explaining the generation of the other teleconnection-pattern (e.g., NAO and PNA) events and examining the mutual relationship between downstream blocking and the phase of NAO or PNA.

On the other hand, the NMI model shows that the negative-over-positive dipole structure of the eddy forcing $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ induced by pre-existing upstream synoptic-scale eddies ($ {\psi '_1} $) is a favorable condition for the initiation of atmospheric blocking when the initial blocking has a large-scale anticyclonic-over-cyclonic dipole structure. While blocking is driven by pre-existing upstream synoptic-scale eddies in the NMI model, the feedback of intensified blocking can lead to eddy deformation. To some extent, the eddy deformation is a result of the feedback of intensified blocking on pre-existing synoptic-scale eddies. The evolution (lifetime, intensity, and movement) of blocking can be significantly influenced by the magnitude of $ {\text{P}}{{\text{V}}_y} $ due to changes in the energy dispersion and nonlinearity of the blocking system and eddy forcing, whereas the zonal movement of blocking is mainly determined by the background westerly wind, the magnitude of $ {\text{P}}{{\text{V}}_y} $ and the blocking amplitude. The NMI model further reveals that the blocking dipole anomaly, synoptic-scale eddies and the background zonal flow tend to possess the same low-frequency timescale during their interaction. These characteristics cannot be revealed by the time-mean eddy-mean flow interaction model used widely in previous theoretical studies.

Because $ {\text{P}}{{\text{V}}_y} $ is an important background factor influencing the evolution of sub-seasonal teleconnection patterns (e.g., blocking, NAO, and PNA), and because the variability of sub-seasonal teleconnection patterns significantly influences weather extremes, it is easy to establish a linkage between climate change and weather extremes through examining the modulation of the AA, AMO, and PDO/IPO on $ {\text{P}}{{\text{V}}_y} $ and the variability of sub-seasonal teleconnection patterns due to the change of $ {\text{P}}{{\text{V}}_y} $. On this basis, a bridge from climate change to changes in teleconnection patterns and associated weather extremes could be well-established by using the NMI model. We believe that using the NMI model to examine the dynamics of atmospheric blocking and other teleconnection patterns and relating such findings to the climate change-induced change will be a new research direction.

However, it should be pointed out that we didn’t discuss the applicability of the NMI model to other types of atmospheric blocking in this paper. In fact, the NMI model can be applied to the cases of the omega-type and wave-train like blockings by considering the different spatial structures of the initial blocking, basic flow, and the pre-existing synoptic-scale eddies. In observation and numerical models, the pre-existing synoptic-scale eddies can allow $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ to have a negative-over-positive or positive-over-negative dipole structure. Only when the eddy forcing $- \nabla \cdot {({{\bf{v}}'_1}{q'_1})_P}$ has a negative-over-positive dipole structure, can it drive the initiation of dipole-type blocking. Such synoptic-scale eddies can also be seen in an internally consistent and more complete model (Frederiksen, 1982). Moreover, the NMI model is based on the quasi-geostrophic PV theory, which ignores the role of ageostrophic motions in blocking development. In future work, the ageostrophic motions should be considered in the NMI model. Of course, the NMI model should also be extended to include the effect of the rapidly varying (< interannual timescales) SST anomaly and moisture processes (Pfahl et al., 2015) on the blocking development in addition to considering the impact of slowly varying (≥ interannual timescales) SST anomalies on $ {\text{P}}{{\text{V}}_{\text{y}}} $. These problems are worthy of further investigation.

Acknowledgements. This research was supported by the National Natural Science Foundation of China (Grant Nos. 42150204 and 42288101) and the Chinese Academy of Sciences Strategic Priority Research Program (Grant No. XDA19070403). The authors would like to thank two anonymous reviewers and Dr. Buwen DONG for their useful suggestions in improving the manuscript.

Author contributions. D. LUO conceptualized the theoretical models, constructed the basic idea of this paper, plotted Figs. 4, 5, 6, 7, and 9, and wrote the manuscript. B. LUO plotted Figs. 2−3 and joined the discussions of this manuscript. W. ZHANG gave some discussions on this manuscript. All the authors contributed to the writing and reviewing of the manuscript.

Data Availability. ERA5 reanalysis data used in this paper was obtained from the ECMWF (https://climate.copernicus.eu/climate-reanalysis).