## 1. Introduction

Atmospheric blocking is a large-scale, quasi-stationary, low-frequency circulation pattern with the lifetime of 10–20 days occurring in midlatitudes (Berggren et al. 1949; Rex 1950; Shukla and Mo 1983). Because the extreme cold spells in winter (Buehler et al. 2011), European temperature extremes in spring (Brunner et al. 2017) and heat waves in summer (Della-Marta et al. 2007; Schaller et al. 2018) are often related to the establishment and maintenance of mid–high-latitude blocking, the formation and maintenance mechanism of atmospheric blocking has been an important research topic in past decades (Yeh 1949; Shutts 1983; Colucci 1985; Haines and Marshall 1987; Holopainen and Fortelius 1987; Luo 2000, 2005; Luo et al. 2014, 2019; Zhang and Luo 2020; Nakamura and Huang 2017, 2018; Aikawa et al. 2019; Paradise et al. 2019).

The numerical experiments have indicated that high-frequency synoptic-scale eddies play a more important role in the generation and maintenance of atmospheric blocking than large-scale topographic forcing (Ji and Tibaldi 1983). Some diagnostic studies have also revealed that the time-mean eddy vorticity flux induced by high-frequency eddies tends to maintain a time-mean blocking anomaly (Shutts 1983; Illari and Marshall 1983; Holopainen and Fortelius 1987; Mullen 1987; Nakamura and Wallace 1993; Aikawa et al. 2019). However, in the previous studies it is difficult to infer the causal relationship between the variation of synoptic-scale eddies and the evolution of blocking from a time-mean perspective, when a time-mean potential vorticity (PV) equation is used (Holopainen and Fortelius 1987). In fact, in the unfiltered daily geopotential height fields the intensified blocking flow behaves as an enhanced meandering of westerly jet streams (Berggren et al. 1949) due to the intensified multiple vortex structure within the blocking region resulting from the northward (southward) displacement of intensified synoptic-scale ridges (deepened troughs) in the upstream side of blocking (Luo 2000, 2005). Such a poleward (equatorward) movement of small-scale ridges (troughs) reflects the presence of cyclonic wave breaking (hereafter CWB) or eddy straining. Thus, in some previous studies the eddy straining or CWB was understood as leading to the blocking formation and maintenance (Shutts 1983; Weijenborg et al. 2012).

Shutts (1983) first attributed the maintenance of dipole blocking to the meridional straining of synoptic-scale eddies in a diffluent basic flow or a stationary blocking. However, from the temporal evolution of the planetary-scale blocking field and synoptic-scale eddies during their interaction in a nonlinear theoretical model of blocking Luo (2000, 2005) demonstrated that the preexisting incident synoptic-scale eddies rather than the eddy straining or CWB are a key driver of the spatiotemporal evolution of blocking and its life cycle (formation, maintenance, and decay). This nonlinear model can describe the life cycle of a blocking event or a strong westerly jet meandering, which is different from the “traffic jam” model of blocking advanced by Nakamura and his collaborators (Nakamura and Huang 2017, 2018; Paradise et al. 2019), because the traffic jam model can only reflect the initial development of blocking as an accumulation of local wave activity rather than its life process. While a dipole blocking can be established by preexisting synoptic-scale eddies (Luo 2000, 2005), the feedback of the intensified blocking has been shown to cause the deformation (straining and merging) of preexisting synoptic-scale eddies or CWB in a nonlinear multiscale interaction (NMI) model (Luo et al. 2014). Thus, Luo et al. (2019) further concluded that the eddy straining or CWB is a result or concomitant phenomenon of the blocking establishment and maintenance rather than a cause. They also found that a small meridional PV gradient (PV_{y}) as a favorable background media tends to maintain blocking through reducing its energy dispersion and strengthening its nonlinearity especially in high latitudes.

However, the previous NMI model is not appropriate for investigating the effect of Arctic amplification (AA) on atmospheric blocking because AA can reduce the meridional temperature gradient (Newson 1973; Cohen et al. 2020) in addition to the change of background westerly wind in mid–high latitudes as well as because this barotropic model ignores the effect of reduced meridional temperature gradient (Luo et al. 2018, 2019). Thus, it is useful to extend our previous barotropic NMI model to include the effect of a three-dimensional background flow. Using this extended NMI model, it becomes feasible to distinguish the different roles of the horizontal (mainly meridional) shear of background westerly wind (BWW) and the magnitude of meridional temperature gradient in the blocking change. This extended NMI model provides a useful theoretical tool for further understanding how AA, continental snow cover, and internal variability influence the evolution of blocking and the generation cause of winter cold extremes.

This paper is organized as follows: In section 2, we describe the blocking–eddy interaction equations in a three-dimensional baroclinic framework as an extended NMI model. In this section, a further simplification of the blocking–eddy interaction equations is also made under some assumptions. The analytical solution of the extended NMI model is presented in section 3. In section 4, we describe theoretical results about how the meridional shear of the BWW and its vertical shear related to the meridional temperature gradient influence the spatiotemporal evolution of eddy-driven blocking wave packet in a barotropic atmosphere without vertical variation and a baroclinic atmosphere with stratification. The main conclusion and discussions are summarized in the final section.

## 2. Model description

We consider a blocking flow governed by a quasigeostrophic baroclinic potential vorticity (PV) equation on a *β* plane in the form

where *ψ*_{T}(*x*, *y*, *z*, *t*) being the total streamfunction; ∇^{2} = ∂^{2}/∂*x*^{2} + ∂^{2}/∂*y*^{2} is the two-dimensional Laplacian operator; *J*(*a*, *b*) = (∂*a*/∂*x*)(∂*b*/∂*y*) − (∂*a*/∂*y*)(∂*b*/∂*x*); *f*_{0} and *β*_{0} are the Coriolis parameter and its meridional gradient at a given reference latitude *φ*_{0}, respectively; *N* is the reference-state Brunt–Väisälä frequency and *ρ*_{0} is the reference state density; *x*, *y*, *z*, and *t* are the zonal, meridional, height, and time coordinates respectively; Γ_{T} is the PV source; and *D*_{T} is the dissipation as the PV sink.

Let us assume that the blocking disturbance is embedded in a basic flow *ψ*_{∞}(*x*, *y*, *z*) in the far field where all the blocking disturbances vanish. In this case, we may assume *ψ*_{T} = *ψ*_{∞}(*x*, *y*, *z*) + *ψ*(*x*, *y*, *z*, *t*), *q*_{T} = *q*_{∞}(*x*, *y*, *z*) + *q*(*x*, *y*, *z*, *t*), Γ_{T} = Γ_{∞} + Γ, and *D*_{T} = *D*_{∞} + *D*, where Γ_{∞} (Γ) and *D*_{∞} (*D*) are the far-field (disturbance) PV source and dissipation, respectively. Under the assumption *J*(*ψ*_{∞}, *q*_{∞}) = Γ_{∞} − *D*_{∞}, from Eq. (1) one can obtain

where

Considering *U* = −∂*ψ*_{∞}/∂*y*, *V* = ∂*ψ*_{∞}/∂*x*, and ∂*U*/∂*x* + ∂*V*/∂*y* = 0, Eq. (2) can be rewritten as

where

Here, we suppose *U*, *V*, PV_{y}, and PV_{x} are slowly varying, then the following nondimensional PV disturbance equation, scaled by the characteristic velocity ^{−1}), length ^{6} m), and height ^{4} m), can be obtained by multiplying Eq. (3) with nondimensional *ϕ*(*z*) and integrating it from 0 to 1 in the vertical direction:

where *ρ*_{0} = *ρ*_{s}*e*^{−z} [*ρ*_{s} is the atmospheric density at Earth’s surface (*z* = 0)] in a nondimensional vertical domain (0, 1) and *N*^{2} is a constant to consider the effect of the horizontal (mainly meridional) and vertical shears of the background westerly wind (BWW) on blocking. As a result, it is easy to obtain PV_{y} = *β* + *V*_{xy} − *U*_{yy} + *F*_{r}*U*_{z} − *F*_{r}*U*_{zz}, PV_{x} = ∇^{2}*V* − *F*_{r}*V*_{z} + *F*_{r}*V*_{zz} and *U*_{yy} = ∂^{2}*U*/∂*y*^{2}, *V*_{xy} = ∂^{2}*V*/∂*x*∂*y*, *U*_{z} = ∂*U*/∂*z*, *U*_{zz} = ∂^{2}*U*/∂*z*^{2}, *V*_{z} = ∂*V*/∂*z*, *V*_{zz} = ∂^{2}*V*/∂*z*^{2},

Since the blocking flow is often driven by synoptic-scale eddies, it is useful to separate the nondimensional disturbance streamfunction *ψ* represents the planetary-scale blocking anomaly with a zonal wavenumber *k* and *j* = 1, 2, …).

As in Luo (2000, 2005) and Luo et al. (2014, 2019), under the zonal scale separation assumption

where *q*(*x*, *y*, *t*) = ∇^{2}*ψ* − *Fψ*, *q*′(*x*, *y*, *t*) = ∇^{2}*ψ*′ − *Fψ*′, and **v**′ = (−∂*ψ*′/∂*y*, ∂*ψ*′/∂*x*) is the horizontal wind vector of synoptic-scale eddies. In the derivation of Eq. (5), *P* of −∇ ⋅ (**v**′*q*′)_{P} denotes that −∇ ⋅ (**v**′*q*′) has the same spatial structure as that of the blocking flow.

In the vertical direction, any nondimensional vertical function *ϕ*(*z*) of the blocking and synoptic-scale eddy components can be expanded as *c*_{0} and *c*_{n} are the constants. When *c*_{n} = 0 and *c*_{0} is a constant, one can have *ϕ*(*z*) = *c*_{0}. In this case, the planetary- and synoptic-scale anomalies possess a barotropic structure without vertical variation. When *c*_{0} ≠ 0, *c*_{1} ≠ 0, and *c*_{n} = 0 for *n* ≥ 2, the blocking anomaly shows a vertical variation, thus reflecting that the blocking and synoptic-scale anomalies can have a baroclinic structure. Crudely speaking, the observed blocking anomaly has a maximum amplitude in the middle and upper tropospheric layers near the 500–300-hPa levels. For this reason, we consider two cases: 1) *c*_{0} = 1 and *c*_{n} = 0 for *n* ≥ 1 (barotropic case) and 2) *c*_{0} = 1 and *c*_{n} = *Z*_{b} ≠ 0 for *n* = 1 (baroclinic structure). On the other hand, because *U* and *V* have horizontal and vertical shears or because PV_{y} and PV_{x} include the horizontal and vertical shears of the background horizontal winds and basic atmospheric stratification, using Eq. (5) can allow us to examine the barotropic and baroclinic effects of the basic-state flow on eddy-driven blocking. Thus, our present model is an extension of the NMI model of eddy-driven blocking wave packet proposed by Luo (2000, 2005) and Luo et al. (2014, 2018, 2019) and Zhang and Luo (2020).

## 3. Analytical solution of the extended NMI model

As used in Luo et al. (2018, 2019) and Zhang and Luo (2020), Eq. (5) can be analytically solved by assuming

Equation (5a) can also be rewritten as ∂*q*/∂*t* + *J*(*ψ*, Θ*q* + PV) + *J*(*ψ*_{∞}, *q*) = −Θ∇ ⋅ (**v**′*q*′)_{P}, where PV = *q*_{∞}(*x*, *y*, *z*) and *ψ*_{∞} is the nondimensional three-dimensional basic-state streamfunction. Under the assumption *J*(*ψ*, Θ*q* + PV) + *J*(*ψ*_{∞}, *q*) ≈ 0 as in Luo et al. (2014) because of approximate linear relationship between *q* and *ψ* in a slowly varying background flow, one can get ∂*q*/∂*t* ≈ −Θ∇ ⋅ (**v**′*q*′)_{P}. Because *t* ~ 0) of blocking, one can easily obtain *q* of the downstream incipient block. This mechanism is the so-called eddy-blocking matching mechanism of blocking proposed by Luo et al. (2014, 2019). The incipient block, as denoted by a preexisting planetary-scale ridge, was also found to be related to the downward propagation of stratospheric planetary waves (Kodera et al. 2013) and other processes. However, it must be emphasized in this theoretical model that the eddy straining or CWB is unimportant for the blocking intensification and decay because

To analytically solve Eq. (5), here we further make some assumptions that *ψ*_{∞}(*x*, *y*, *z*) is slowly varying in space compared to the blocking anomaly and the meridional background wind *V* is much weaker than the background zonal wind *U* as observed in Luo et al. (2018). Using the Wentzel–Kramers–Brillouin (WKB) method as in Luo (2005), the approximate analytical solution of the nondimensional total streamfunction field *ψ*_{T} of blocking from Eq. (5) in a *β*-plane channel with a nondimensional width of *L*_{y} can be obtained in fast variable form of

where *ω* = *Uk* − (PV_{y}*k*)/(*k*^{2} + *m*^{2} + *F*), *j* = 1, 2), *α*_{1} = 1, *α*_{2} = *α* = −1, *C*_{g} = ∂*ω*/∂*k* = *U* − [PV_{y}(*m*^{2} + *F* − *k*^{2})]/(*k*^{2} + *m*^{2} + *F*)^{2}, *ω* of the blocking carrier wave, which reflects the forcing of the blocking dipole by preexisting synoptic-scale eddies, and cc denotes the complex conjugate of its preceding term. One can have *ω* ~ 0 in Eq. (6h). Also note that *B* of a nonlinear blocking wave packet, *λ* = [3(*m*^{2} + *F*) − *k*^{2}]PV_{y}*k*/(*k*^{2} + *m*^{2} + *F*)^{3} is the linear dispersion strength, *δ* = Θ^{2}*δ*_{B} is the nonlinearity strength, *δ*_{B} = *δ*_{N}/PV_{y}, *G* = Θ*G*_{B}, *q*_{n} = *q*_{Nn}/PV_{y}, *m* = −2*π*/*L*_{y}, *k* = 2*k*_{0}, *k*_{0} = 1/(6.371 cos*φ*_{0}), *j* = 1, 2), *n*_{j} is the positive integer (e.g., *n*_{1} = 9 and *n*_{2} = 11 used in this paper), *x* = −*x*_{T}, *ε* is a small parameter, *a*_{0} is the constant maximum eddy intensity, and *μ* > 0. Clearly, the linear dispersion strength is proportional to the background meridional PV gradient (PV_{y}), whereas the nonlinearity strength is proportional to the inverse of PV_{y}. Thus, there is an inverse ratio rule of the linear dispersion–nonlinearity relation for a blocking event in a three-dimensional baroclinic flow. Here, *p*_{j}, *r*_{j}, *s*_{j}, *h*_{j,} and *Q*_{j} in Eq. (6g) are given in appendix A, whereas the coefficients *δ*_{N}, *q*_{Nn}, *g*_{n}, and *G*_{B} of the amplitude Eq. (6h) of blocking wave packet are defined in appendix B.

In Eq. (6), the total streamfunction field *ψ*_{T} of eddy-driven blocking flow consists of planetary- and synoptic-scale parts (*ψ*_{P} and *ψ*_{P} represents the planetary-scale field of blocking, and *ϕ*(*z*) during the blocking life cycle. In *ψ*_{P}, *ψ*_{B} is the blocking wavy anomaly with the removal of *ψ*_{∞}(*x*, *y*, *z*) and *ψ*_{m} from *ψ*_{P}, whereas *ψ*_{m} represents the change of the mean zonal wind due to the feedback of intensified blocking wavy anomaly *ψ*_{B} during the blocking life cycle. Since *ϕ*(*z*) and *ψ*_{B} because *B*, whereas

Since the preexisting eddy forcing *ψ*_{B}, is also considered as the evolution of a nonlinear wave packet in the form of *B* exp[*i*(*kx* − *ωt*)], which is described by a nonlinear Schrödinger (NLS) equation with a preexisting eddy forcing (Luo 2000, 2005; Luo et al. 2014, 2019; Zhang and Luo 2020).

Here, we have extended our previous nonlinear wave packet theory to include a general three-dimensional basic flow to examine the effect of the basic flow with horizontal and vertical shears on eddy-driven blocking, even though the evolution of blocking depends on the initial value of preexisting planetary-scale wave and synoptic-scale eddies. As an example, we consider a barotropic case and show a schematic diagram in Fig. 1 to depict how preexisting upstream synoptic-scale eddies create a blocking from an incipient block through the forcing of

Values of given parameters in the extended NMI model.

When *c*_{0} = 1 and *c*_{n} = 0 for *n* ≥ 1 are allowed, there are *ϕ*(*z*) = 1 and Θ = 1. In this case, we have *F* = 0. However, when there is *ϕ*(*z*) = 1 + *Z*_{b} sin*πz* for *c*_{0} = 1, *c*_{1} = *Z*_{b} and *c*_{n} = 0 (*n* ≥ 2), we can obtain *F* = 0 and Θ = 1 is found for *Z*_{b} = 0 (a barotropic case with no vertical variation). In this case, we further have PV_{y} = *β* + *V*_{xy} − *U*_{yy} and PV_{x} = ∇^{2}*V* for a purely barotropic background horizontal wind with *U* = *U*(*x*, *y*), *V* = *V*(*x*, *y*), and *ϕ*(*z*) = 1. As a result, Eq. (6) reduces to the NMI model in a zonally varying barotropic basic flow presented by Luo et al. (2019) and Zhang and Luo (2020). For *Z*_{b} > 0, there is Θ > 1. For example, we have Θ = 1.13 for *Z*_{b} = 0.2. This reflects that the presence of Θ > 1 tends to enhance the nonlinearity of blocking system due to *δ* = Θ^{2}*δ*_{B} > *δ*_{B} and strengthen the preexisting eddy forcing because of *G* = Θ*G*_{B} > *G*_{B}. Thus, in some degree the baroclinic structure of blocking tends to favor the maintenance itself because of increased nonlinearity. Because PV_{y} includes the horizontal and vertical shears of the background flow, the nonlinear behavior (duration, strength, and movement) of eddy-driven blocking is inevitably influenced by the horizontal and vertical shears of background basic flow.

It is useful to divide PV_{y} into _{y} is small, the linear dispersion *λ* is weak and the nonlinearity strength *δ* is strong in the nonlinear wave packet Eq. (6h), thus indicating that the blocking can last for a long time. When one assumes *U* = *U*_{B}(*x*, *y*) + *U*_{T}(*z*) and *V* = *V*_{B}(*x*, *y*) + *V*_{T}(*z*), where *U*_{B}(*x*, *y*) or *V*_{B}(*x*, *y*) and *U*_{T}(*z*) or *V*_{T}(*z*) are the barotropic and baroclinic parts of the basic flow, respectively, it is easy to obtain

When the background wind is purely zonal and *U* = *u*_{0} is a constant, one can have PV_{y} = *β*. The mathematical form of PV_{y} is different from the result of PV_{y} = *β* + *Fu*_{0} in a uniform westerly wind *u*_{0} obtained by Luo et al. (2019) and Zhang and Luo (2020) when the divergence of horizontal winds is considered in a barotropic model as in Yeh (1949). Although the meridional and vertical variations of *B* in *y* and *z* directions are not explicitly included in Eq. (6h), it essentially shows meridional and vertical variations in that PV_{y} and *U* have been considered as being the functions of *y* and *z*. For given initial blocking and parameter conditions, the high-order split-step Fourier scheme (Muslu and Erbay 2005) is used to solve Eq. (6h) to derive the spatiotemporal solution of the envelope amplitude *B* of the blocking wave packet. In the split-step Fourier scheme calculation of Eq. (6h), the blocking wave packet has been assumed to be a superposition of many linear modes or waves (Luo et al. 2019). Moreover, we also use the periodic boundary conditions same as used in Luo et al. (2019).

In this paper, we consider a BWW *U* = *U*(*x*, *y*, *z*) and *V* ≈ 0 as a pure zonal basic flow. We refer to *U* = *U*_{B}(*x*, *y*) as a barotropic BWW if *U* = *U*_{B}(*x*, *y*) and *U*_{T}(*z*) = 0 are assumed. Correspondingly, *U* = *U*_{T}(*z*) is referred to as a baroclinic BWW if *U*_{B}(*x*, *y*) = 0 is assumed. Because *ϕ*(*z*) = 1 + *Z*_{b} sin*πz*, the eddy-driven blocking may be called a “baroclinic (barotropic) blocking” with (without) vertical variation for *Z*_{b} > 0 (*Z*_{b} = 0). It is also noted that the barotropic blocking becomes a baroclinic blocking even under *Z*_{b} = 0 and *F* = 0, if the effect of *U*_{z} for *F*_{r} ≠ 0. Thus, the evolution (duration, movement and strength) of blocking depends not only on the horizontal (mainly meridional) shear of the BWW (the value of *F*_{r} ≠ 0 (the value of *F* ≠ 0 in a baroclinic atmosphere with stratification. This formula shows that the frequency Δ*ω* of the preexisting eddy forcing is smaller in a baroclinic blocking flow (*F* ≠ 0) than that in a barotropic blocking flow [*F* = 0 and *U* = *U*_{B}(*x*, *y*) for *U*_{T}(*z*) = 0]. Thus, the preexisting eddy forcing as a driver of blocking has inevitably a longer duration for a baroclinic blocking flow than for a barotropic blocking flow, leading to a long-lived baroclinic blocking.

## 4. Theoretical results

While Eq. (6) is the analytical solution of eddy-driven blocking flow, it is useful to present the spatiotemporal evolution solution of the blocking field in this section. In our calculations, we choose the parameters *B*(*x*, *y*, *z*, 0) = 0.3, *Z*_{b} = 0, and *Z*_{b} = 0.2, but allow the BWW to vary. The other fixed parameters *n*_{1} = 9, *n*_{2} = 11, *L*_{y} = 5, *N*^{2} = 2 × 10^{−4} s^{−1}, *μ* = 1.2, *ε* = 0.24, *x*_{T} = 2.87/2, *a*_{0} = 0.12, and *φ*_{0} = 55°N are shown in Table 1. For *Z*_{b} = 0.2, we have *F* = 0.82. We also test the sensitivity of the obtained results to the background flow parameters. The obtained results are similar (not shown).

To see whether the background media where a blocking occurs changes with the atmospheric stratification, we show the variations of the linear phase speed *C*_{p}, linear group velocity *C*_{g}, linear dispersion term *λ*, nonlinearity strength *δ*, the eddy forcing frequency Δ*ω*, eddy forcing coefficient |*G*|, *δ*/*λ*_{,} and *C*_{gp} = *C*_{g} − *C*_{p} with the parameter *F* in Fig. 2 for a uniform BWW with *U* = 0.7, when *F* is considered as a changing parameter. It is found that while *C*_{p} (Fig. 2a) and *C*_{g} (Fig. 2b) increase with the increasing of *F*, *C*_{gp} (Fig. 2h) decreases with the increasing of *F*, thus indicating that a weak energy dispersion can be seen in a large value range of *F*. Because *δ*/*λ* measures a relative magnitude between the nonlinearity strength *δ* and linear dispersion term *λ*, the value of *δ*/*λ* can reflect whether the value of *F* changes the dispersion or nonlinearity of blocking system. We also see that *δ*/*λ* increases with the increasing of *F* (Fig. 2g), though both *λ* (Fig. 2c) and *δ* (Fig. 2d) decrease with *F* and *λ* has a larger decrease than *δ*. Thus, it is concluded that a large value of *F* favors the weak dispersion or strong nonlinearity of blocking.

On the other hand, because Δ*ω* decreases with the increasing of *F*, the preexisting eddy forcing has a long duration in the region of large *F*. Thus, eddy-driven blocking is inevitably long-lived in a large *F* region, which can also be explained by *C*_{gp} = 2*k*^{2}PV_{y}/(*k*^{2} + *m*^{2} + *F*)^{2} obtained from the linear theory. When PV_{y} is small or when *F* is large, *C*_{gp} is small. In this case, the energy dispersion of the linear Rossby wave packet is weak. The dispersion time scale of the Rossby wave packet is *T*_{β} = 2*π*/*C*_{gp} ~ (*k*^{2} + *m*^{2} + *F*)^{2}/(PV_{y}*k*^{2}), which does not require that the zonal wavelength of synoptic-scale eddies must be small. Instead, it depends on the zonal wavelength of Rossby wave packet and the magnitude of background PV_{y}. When *k* or PV_{y} is small, *T*_{β} is large. As a result, the linear wave packet has a long dispersion time scale, corresponding to a weak energy dispersion. Naturally, the longer zonal scale of the Rossby wave packet and small PV_{y} tend to maintain the wave packet itself. This theory is different from that of Hoskins and James (2014), who concluded that the length scale of the synoptic-scale eddies must be small to allow the cascade of energy to larger scales. But in our nonlinear theory as an extension of the linear wave theory, such a condition is unnecessary because the extended NMI model considers blocking as a nonlinear initial value problem. Instead, only

In this paper, we fix the zonal wavelength of the initial blocking to examine the likely effect of PV_{y} and its *U*_{B}(*x*, *y*) and baroclinic *U*_{T}(*z*) flows to examine their respective impact on eddy-driven blocking.

### a. Temporal evolution of barotropic blocking

Here, we consider a barotropic case [*ϕ*(*z*) = 1, *Z*_{b} = 0, *U*_{T}(*z*) = 0 and *F* = 0] as we have noted above. We also assume *γ* > 0 and *u*_{0} is a constant that represents the uniform part of the barotropic BWW, with Δ*u* being its horizontal shear, and *x*_{0} and *y*_{0} are the zonal and meridional positions of the maximum shear of the BWW. For *u*_{0} = 0.7, *γ* = 0.3, *y*_{0} = 3.75, *x*_{0} = 0, we show the horizontal distributions of *U*(*x*, *y*) and associated *u* = −0.2. This figure shows that there are smaller barotropic BWW and PV_{y} in higher latitudes for Δ*u* = −0.2. In contrast, larger barotropic BWW and PV_{y} are seen in higher latitudes for Δ*u* = 0.2 (not shown). Correspondingly, the meridional profiles of *U*(*x*, *y*) and associated PV_{y} at *x* = 0 are shown in Figs. 3c and 3d for Δ*u* = −0.2 (dashed line), Δ*u* = 0 (dot–dashed line), and Δ*u* = 0.2 (solid line). It is also found that in the high latitudes, PV_{y} decreases from 1.47 to 1.15 (Fig. 3d) when the barotropic BWW strength decreases from 0.9 to 0.5 along *y* = 3.75 (Fig. 3c). In this section, we examine how such barotropic BWW and PV_{y} distributions affect the evolution of eddy-driven blocking. Below we only present the results for *γ* = 0.3 and

Figure 4 shows the variations of *C*_{p}, *C*_{g}, *λ*, *δ*, Δ*ω*, |*G*|, *δ*/*λ*, and *C*_{gp} with the latitude and the value of *F* for *u*_{0} = 0.7, *γ* = 0.3, *y*_{0} = 3.75, and Δ*u* = −0.2. Clearly, when the barotropic BWW is small in higher latitudes, *δ*/*λ* is large (Fig. 4g) and *C*_{gp} and Δ*ω* are small (Figs. 4h,e). Thus, a weaker PV_{y} in higher latitudes related to a smaller BWW as a weaker dispersion or a stronger nonlinearity media may provide a favorable environment for the longer lifetime of blocking in a certain range of *u*_{0}. We also examine the sensitivity of the blocking evolution to the value of *u*_{0}.

Using the same parameters as in Table 1, we show instantaneous fields of planetary-scale streamfunction *ψ*_{P}, blocking wavy anomaly *ψ*_{B}, synoptic-scale eddy streamfunction *ψ*_{T} *ϕ*(*z*) = 1] during its life cycle in Fig. 5 for a uniform BWW with *U* = *u*_{0} = 0.7. It is noted that initial synoptic-scale eddies *x* = 0 (Fig. 5a for day 0). Through the forcing of *B* of blocking.

When the blocking disappears or when *B* = 0, *ψ*_{T} (Fig. 5d), the establishment of eddy-driven blocking is characterized by a large meandering of westerly jet streams, which is composed of several isolated synoptic-scale cyclonic and anticyclonic vortices within the blocking region. A large meandering of the westerly jet streams also reflects the presence of CWB or eddy straining related to the northward (southward) displacement of intensified synoptic-scale ridges (troughs) in the blocking region and its upstream side.

Here, we further examine how reduced barotropic BWW and PV_{y} in higher latitudes influence the evolution of eddy-driven barotropic blocking. For *u*_{0} = 0.7, Δ*u* = −0.2, *γ* = 0.3, and *y*_{0} = 3.75, we show the instantaneous *ψ*_{P}, *ψ*_{B}, *ψ*_{T} fields of eddy-driven barotropic blocking [*ϕ*(*z*) = 1] in Fig. 6 during its evolution process. It is interesting to note that the barotropic blocking shows a northwest–southeast (NW–SE)-oriented dipole due to the presence of small barotropic BWW and PV_{y} in higher latitudes, which undergoes an enhanced retrogression and is further intensified (Figs. 6a,b) relative to that in a uniform BWW (Figs. 5a,b). The lifetime of this barotropic blocking is slightly lengthened as well. In Fig. 6c, we can see a significant horizontal straining of synoptic-scale eddies due to the meridional shear of the barotropic BWW with a smaller strength in higher latitudes than in lower latitudes (Fig. 3c) and due to the feedback of intensified NW–SE-oriented blocking dipole (Fig. 6b). The strained synoptic-scale eddies move eastward more rapidly over the south side of the blocking region than over its north side (Fig. 6c) so that the spatial pattern of the westerly jet stream meandering during the blocking life cycle (Fig. 6d) is slightly different from that in a uniform BWW (Fig. 5d).

To see the impact of the barotropic BWW and PV_{y} on the lifetime and local persistence of eddy-driven dipole blocking, the maximum amplitude *ψ*_{Max} of the anticyclonic anomaly of blocking wavy part *ψ*_{B} at *y* = 3.75 for each day is defined as the daily strength of dipole blocking to characterize the lifetime of blocking. It is also useful to consider the value of the domain-averaged blocking wavy anomaly streamfunction *ψ*_{A} over −1.5 ≤ *x* ≤ −0.5 and 3.5 ≤ *y* ≤ 4 at every day during the blocking life cycle as an daily index reflecting the local persistence of blocking because the eddy-driven blocking often moves upstream in a weak BWW. We show the time–longitude evolution of the blocking wavy anomaly *ψ*_{B} at *y* = 3.75 as reflecting the propagation or movement of blocking wave packet and the time series of daily *ψ*_{Max} and *ψ*_{A} during the blocking life cycle in Fig. 7 for *u*_{0} = 0.7 and *u*_{0} = 0.5 of *γ* = 0.3, and *y*_{0} = 3.75 with three cases: Δ*u* = −0.2, Δ*u* = 0, and Δ*u* = 0.2.

It is seen that the lifetime, strength, and movement of eddy-driven barotropic blocking depend on the strength of the barotropic BWW and its spatial distribution (Figs. 7a–f). Figures 7a–c (Figs. 7d–f) show the case of *u*_{0} = 0.7 (*u*_{0} = 0.5). In general, the lifetime of the barotropic blocking is less than 15 days (thick black lines in Figs. 7a–c,e,f), unless the barotropic BWW is particularly small in higher latitudes (Fig. 7d). When the BWW is stronger in higher latitudes, the barotropic blocking moves eastward (Fig. 7c) due to large high-latitude PV_{y}. In contrast, the eastward movement of blocking (Fig. 7b) is suppressed and even its slow retrogression can be seen (Fig. 7a), when *U* or PV_{y} is small in higher latitudes. Specifically, when the uniform part of *U* (e.g., *u*_{0} = 0.5) is smaller (Fig. 7d), the barotropic blocking shows more notable retrogression, slower decay, and has longer lifetime, while its life period is slightly short for Δ*u* = 0 (Fig. 7e) or Δ*u* = 0.2 (Fig. 7f) compared to that of *u*_{0} = 0.7 for Δ*u* = 0 (Fig. 7b) or Δ*u* = 0.2 (Fig. 7c). The movement of eddy-driven blocking as a wave packet propagation can be explained by using the nonlinear phase speed of

As can be seen in Figs. 7g–j, while the lifetime and strength of eddy-driven barotropic blocking is slightly influenced by the strength of *U* or PV_{y} in higher latitudes (Fig. 7g), its local persistence seems to depend more strongly on the magnitude and meridional distribution of *U* or PV_{y} (Fig. 7h). When the uniform part of *U* is particularly small, a long-lived blocking with large retrogression and slow decay is easily seen (Fig. 7i). For this case, the difference of the local persistence of blocking becomes less distinct among Δ*u* = −0.2, Δ*u* = 0, and Δ*u* = 0.2 (Fig. 7j), in which the local persistence of blocking is shortest for Δ*u* = −0.2. The above results are obtained based on a pure barotropic case (*Z*_{b} = 0, *F* = 0, and

It needs to be pointed out that the barotropic blocking becomes a baroclinic case even for *Z*_{b} = 0 and *F* = 0, if there are *Z*_{b} ≠ 0 and *F* ≠ 0. These problems are further examined in the next subsection.

### b. Effect of barotropic background westerly wind on the evolution of baroclinic blocking in the baroclinic atmosphere with stratification

In a baroclinic atmosphere with *Z*_{b} = 0.2 and *F* = 0.82 we show the *ψ*_{P}, *ψ*_{B}, *ψ*_{T} fields of eddy-driven baroclinic blocking at *z* = 0 during its life cycle in Fig. 8 for Δ*u* = 0 and Δ*u* = −0.2 of *u*_{0} = 0.7, *γ* = 0.3, and *y*_{0} = 3.75. It is found that for Δ*u* = 0 (a uniform barotropic BBW) the baroclinic blocking shows an antisymmetric dipole, long lifetime and notable eastward movement especially during the blocking decaying phase (Figs. 8a,b), whereas it shows a NW–SE orientation and a reduced eastward movement for Δ*u* = −0.2 (Figs. 8c,d). A large increase in the local persistence of blocking is also seen in this case. For the same conditions as in Figs. 8c and 8d, the eddy-driven blocking can have large amplitude in the midtroposphere (*z* = 0.5) as shown in Fig. 9a. We note that the blocking amplitude increases with height and then decreases with height above the midtroposphere, as seen from a comparison between Figs. 9a and 8c. The evolving synoptic-scale eddies

Figure 10 shows the result of Δ*u* = 0.2 in the midtroposphere for the same parameters as in Fig. 9. It is seen that the eddy-driven blocking exhibits a northeast–southwest (NE–SW) orientated dipole structure and remains rather strong even after day 18 (Figs. 10a,b) because the barotropic BWW or PV_{y} is stronger in higher latitudes than in lower latitudes (solid lines in Figs. 3c and 3d). In addition, because the synoptic-scale eddies move eastward more rapidly in higher latitudes than in lower latitudes (Fig. 10c), the spatial pattern of the meandering westerly jet stream (Fig. 10d) is different from that of Δ*u* = −0.2 as seen in Fig. 9d. Thus, while the local persistence of eddy-driven blocking is weak for Δ*u* = 0.2, it has long lifetime, fast eastward movement, and slow decay (Figs. 10a,b).

For *γ* = 0.3, *y*_{0} = 3.75, *Z*_{b} = 0.2, and *F* = 0.82 with three cases of Δ*u* = −0.2, Δ*u* = 0, and Δ*u* = 0.2, we show the time–longitude evolution of the blocking wavy anomaly *ψ*_{B} at *y* = 3.75 and the time series of daily *ψ*_{Max} and *ψ*_{A} of eddy-driven baroclinic blocking during the blocking life cycle in Fig. 11 for *u*_{0} = 0.7 and *u*_{0} = 0.5. It is found that the eddy-driven baroclinic blocking can have longer lifetime in the baroclinic atmosphere with stratification (*Z*_{b} = 0.2 and *F* = 0.82) (Figs. 11a–f) than in the pure barotropic atmosphere (*Z*_{b} = 0 and *F* = 0) (Figs. 7a–c,e–f), even when the barotropic BWW is less weak. The evolution (movement, lifetime, strength) of blocking is also found to depend on the magnitude of *u*_{0} or *U* and *U* and *u*_{0} = 0.7, the slow decay of this baroclinic blocking is stronger for Δ*u* = 0.2 (solid line in Fig. 11g) than that for Δ*u* = −0.2 (dot–dashed line in Fig. 11g) or Δ*u* = 0 (dashed line in Fig. 11g).

In addition, the value of the uniform part *u*_{0} of the barotropic BWW can affect the lifetime, movement, and local persistence of eddy-driven baroclinic blocking. It is noted that *u*_{0} = 0.7 and *u*_{0} = 0.5 for each case of Δ*u* = 0.2, Δ*u* = −0.2, and Δ*u* = 0. While the lifetime of blocking becomes shorter for *u*_{0} = 0.5 than for *u*_{0} = 0.7 under the Δ*u* = 0 or Δ*u* = 0.2 condition, its local persistence is increased to possess a long local duration (Figs. 11h,j) and a less eastward movement (Figs. 11e,f). Although *u*_{0} = 0.5 and *u*_{0} = 0.7 for Δ*u* = −0.2, the lifetime of blocking is slightly longer and its retrogression becomes more evident for *u*_{0} = 0.5 (Figs. 11d,i) than for *u*_{0} = 0.7 (Figs. 11a,g). Moreover, we also find that the slow decay of eddy-driven blocking in the baroclinic atmosphere with stratification is less distinct than that of eddy-driven barotropic blocking (Fig. 7i) under the weak BWW or weak *γ* = 0.3, *y*_{0} = 3.75, *u*_{0} = 0.5, and Δ*u* = −0.2. The above results clearly reveal that the eddy-driven blocking tends to have larger amplitude in the midtroposphere and longer lifetime in a baroclinic atmosphere with stratification than in a pure barotropic atmosphere (no vertical variation), whereas the movement and local persistence of blocking can be largely influenced by the magnitude of the barotropic BWW and

### c. Impact of baroclinic background westerly wind or ${\text{PV}}_{\mathit{y}}^{\mathit{T}}$ on eddy-driven blocking

Here, we further consider a pure baroclinic BWW as *ν* = 2 and *z*_{0} = 0.5. For *u*_{0} = 0.7, we show the vertical profiles of *U*(*z*), *U*_{z}, −*U*_{zz}, and PV_{y} = *β* − *F*_{r}*U*_{zz} + *F*_{r}*U*_{z} (*F*_{r} = 0.72 in Table 1) in Fig. 12 for Δ*u* = 0.2, Δ*u* = 0, and Δ*u* = −0.2. In this figure, the baroclinic BWW has no horizontal shear. It is noted that while *U*(*z*) and −*U*_{zz} have a highest or lowest value at *z* = 0.5 and *U*_{z} is zero at *z* = 0.5 (Figs. 12a–c), PV_{y} tends to have a maximum (minimum) amplitude slightly below *z* = 0.5 (Fig. 12d) due to the presence of *U*_{z} (Fig. 12b) for Δ*u* = 0.2 (Δ*u* = −0.2). We also note that *U*_{z} (Fig. 12b) is much smaller than −*U*_{zz} (Fig. 12c), thus indicating that −*U*_{zz} rather than *U*_{z} plays a major role in the PV_{y} change. In other words, the vertical variation of MTG with height contributes more importantly to the PV_{y} change than the MTG itself, because *U*_{z} represents the value of the MTG. We further find that the change of *u* = −0.2 and Δ*u* = 0.2 is larger than that of

As noted above, an eddy-driven blocking can be influenced by PV_{y} = *β* − *F*_{r}*U*_{zz} + *F*_{r}*U*_{z} even for *Z*_{b} = 0 and *F* = 0, when the BWW is a pure baroclinic flow. Here, we first examine this case. We show the planetary-scale streamfunction *ψ*_{P} field at *z* = 0 of eddy-driven blocking for *Z*_{b} = 0 and *F* = 0 in a baroclinic BWW of *u*_{0} = 0.7, *ν* = 2, and *z*_{0} = 0.5 in Figs. 13a and 13b for Δ*u* = −0.2 and Δ*u* = 0.2. The corresponding time–longitude evolution of the blocking wavy anomaly *ψ*_{P} at *y* = 3.75 during the blocking life cycle is shown in Figs. 13c and 13d. It is clearly found that a small PV_{y} or *u* = −0.2. In contrast, a large PV_{y} or *u* = 0.2 largely reduces the blocking’s strength (Fig. 13b), shortens its lifetime and promotes its eastward movement (Fig. 13d). Such a dependence of the lifetime, local persistence, and strength of blocking on the magnitude of baroclinic BWW or *ψ*_{Max} and *ψ*_{A} as shown in Figs. 13e and 13f. Clearly, less local persistence of blocking is evident for Δ*u* = 0.2 in Fig. 13f.

For Δ*u* = −0.2 or a small *z* = 0.5) than in Earth’s surface (*z* = 0) (Fig. 14a). In contrast, the blocking has shorter lifetime and is much weaker in the midtroposphere than in Earth’s surface for Δ*u* = 0.2 or a large *U*(*z*), *U*_{z}, −*U*_{zz}, and PV_{y} = *β* − *F*_{r}*U*_{zz} + *F*_{r}*U*_{z} with height for *u*_{0} = 0.7, *ν* = 3, *z*_{0} = 0.75, and Δ*u* = 0.2 (solid line), *U* = *u*_{0} with *u*_{0} = 0.7 (dashed line) and *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2) with *u*_{0} = 0.7 and Δ*u* = −0.2 (dot–dashed line). It is seen that while the baroclinic BWW (Fig. 15a) is weaker for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2) in the lower troposphere than in the upper troposphere, it is stronger (weaker) for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2)] than *U* = *u*_{0}. In addition, we note that PV_{y} is smaller in the lower troposphere for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2) (Fig. 15d), but larger for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2). For the two types of baroclinic BBWs, we show the time–longitude evolution of the blocking wavy anomaly *ψ*_{P} at *y* = 3.75 and *z* = 0 during the blocking life cycle and the time series of their corresponding daily *ψ*_{Max} and *ψ*_{A} in Fig. 16 for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2). It is also found that when PV_{y} is small in the lower troposphere, the eddy-driven blocking can have longer lifetime and larger strength (Figs. 16a,b) than for *U* = 0.7 (Fig. 7b), even though the BWW is weaker for *U* = 0.7 than for _{y} in the lower troposphere is favorable for the long lifetime and large strength of eddy-driven blocking. However, we find that the movement of eddy-driven blocking depends on the strength and vertical profile of the baroclinic BWW. While the blocking moves eastward for *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2). In this case, *U*(*z*) = *u*_{0} + Δ*u* cos(*πz*/2) _{y} in the lower to midtroposphere (not shown).

We further repeat the same calculation as in Fig. 13 for *Z*_{b} = 0.2 and *F* = 0.82, and show the calculation results in Fig. 17. Clearly, the eddy-driven blocking has longer lifetime in a baroclinic atmosphere with stratification (*Z*_{b} = 0.2 and *F* = 0.82) (Figs. 17a,b) than in a barotropic atmosphere without the effect of stratification (*Z*_{b} = 0 and *F* = 0) (Figs. 13a,b), even though the PV_{y} field includes the effect of stratification (*F*_{r} ≠ 0). It is further found that while the blocking has larger strength (Fig. 17a) and less eastward movement (Fig. 17c) for a small *u* = −0.2 than for a large *u* = 0.2 (Figs. 17b,d), it shows slower decay for a large *u* = 0.2 (solid line in Fig. 17e). Moreover, the increased local persistence of eddy-driven blocking is more evident for a small *u* = −0.2 than for a large *u* = 0.2 (Fig. 17f), though the blocking has long lifetime for Δ*u* = −0.2 and Δ*u* = 0.2 (Figs. 17c,d).

We also calculate a weak baroclinic BWW case with *u*_{0} = 0.5 for the same parameters as in Fig. 17 and show the results in Fig. 18. The results similar to those in Fig. 17 are found for *u*_{0} = 0.5, but the slow decay of blocking is almost invisible for a large *u* = 0.2. Specifically, it is found that the eddy-driven blocking has longer lifetime, larger strength and stronger local persistence for a small *u* = −0.2 (Figs. 18a,c,e,f) than for a large *u* = 0.2 (Figs. 18b,d,e,f). Overall, the weak baroclinic BWW and small

Since *u* = −0.2 represents the effect of reduced MTG, it is thought that the reduced MTG favors increased local persistence and large amplitude of blocking or a persistent meandering of westerly jet streams. Naturally, our nonlinear multiscale theory here can apply to the effects of AA (or sea ice decline), Eurasian snow cover, and internal low-frequency variability on the winter midlatitude blocking and cold extremes in terms of the MTG and zonal wind changes (Newson 1973; Luo et al. 2018; Cohen et al. 2020; Li and Luo 2019). Further study along this direction will be reported in another paper.

## 5. Conclusions and discussion

In this paper, the nonlinear multiscale interaction (NMI) model of atmospheric blocking proposed by Luo (2000, 2005) and Luo et al. (2014, 2019) and Zhang and Luo (2020) has been extended to include a three-dimensional background flow to examine how the horizontal or vertical shear of a three-dimensional background flow influences the spatiotemporal evolution of eddy-driven blocking. In the extended NMI model, the eddy-driven blocking is considered as a nonlinear Rossby wave packet governed by a forced NLS equation (Luo 2000, 2005), which is initiated by preexisting synoptic-scale eddies upstream as a nonlinear initial-value problem as in Luo et al. (2019). While this model emphasizes the key role of preexisting incident synoptic-scale eddies in the life cycle of blocking, it considers the eddy straining or CWB as a result of the feedback of intensified blocking on preexisting synoptic-scale eddies.

In this model, the spatiotemporal evolution of eddy-driven blocking does not only depend on the parameters of the incipient block and preexisting synoptic-scale eddies, but also on the background field. Although the basic winds and meridional temperature gradient (MTG) can be considered as a background field, the two parameters were not unified into a single factor influencing the evolution of blocking in the previous models. In the present paper, we have unified the background wind and MTG into the background meridional PV gradient (PV_{y}) as a single factor affecting the temporal evolution of blocking. Because the lifetime, movement and strength of blocking are significantly influenced by the magnitude of PV_{y}, one can quantify the different roles of the dynamical PV gradient _{y} into

On the other hand, because the energy dispersion, nonlinearity strength and preexisting eddy forcing in the blocking wave packet equation depend on the values of *F*, *Z*_{b} = 0 and *F* = 0) the eddy-driven barotropic blocking has a short lifetime less than 15 days, whose spatial pattern or horizontal orientation is dominated by the magnitude and meridional distribution of

In a baroclinic atmosphere with stratification (e.g., *Z*_{b} = 0.2 and *F* = 0.82), the eddy-driven baroclinic blocking tends to have long lifetime near 20 days and large amplitude in the midtroposphere even under the vanishing of

We also examined the impact of baroclinic BWW or _{y} is larger than its *U*_{z}) Thus, the magnitude of PV_{y} including *U _{z}* and

*U*is an appropriate parameter describing the change of blocking.

_{zz}Moreover, it is worthy of noting that our theory here does not apply to the real case. For example, how a changing climate (i.e., changes in AA and SST anomalies in the North Pacific and North Atlantic) influences the blocking variability through changing

## Acknowledgments

This research was supported by the National Key Research and Development Program of China (2016YFA0601802), the Chinese Academy of Sciences Strategic Priority Research Program (Grant XDA19070403), and the National Science Foundation of China (Grants 41790473 and 41430533).

## APPENDIX A

### Coefficients of Deformed Eddies in Eq. (6g) in the Extended NMI Model

The coefficients of deformed eddies in Eq. (6g) are defined as

## APPENDIX B

### Coefficients of the Blocking Wave Packet Eq. (6h) in the Extended NMI Model

The following equations define the coefficients of the nonlinearity and eddy forcing terms of the blocking wave packet Eq. (6h):

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