## 1. Introduction

The general circulation of Earth’s extratropical troposphere is dominated by traveling weather systems (ridges and troughs) embedded in the westerly winds of the jet stream. Occasionally the jet stream develops an anomalous, persistent meandering in a certain region and disrupts the normal eastward migration of these weather systems: a condition known as blocking. The stalled ridges and troughs cause heat waves, droughts, prolonged rain, and other abnormal weather patterns. Due to its association with climate extremes, blocking and its response to climate change have been studied extensively (e.g., Woollings et al. 2018, and references therein). Despite its impact on society, blocking remains a challenging problem in numerical weather prediction (Pelly and Hoskins 2003; Jia et al. 2014) and particularly in future climate projections (Woollings et al. 2018). In fact, a precise definition of a block itself is somewhat elusive because of its multifaceted characteristics. Many indices have been proposed, and while most produce consistent climatologies, they often disagree in identifying individual events and in evaluating the effects of climate change on the statistics of blocking (Barnes et al. 2012, 2014).

The difficulty stems partly from the lack of definitive theory for the onset of persistent jet anomalies. Proposed theories for block formation and maintenance include resonance between stationary Rossby waves and forcing (Charney and DeVore 1979; Tung and Lindzen 1979; Brunet 1994; Petoukhov et al. 2013), modon in a shear flow (McWilliams 1980; Malguzzi and Malanotte-Rizzoli 1984; Haines and Marshall 1987; Butchart et al. 1989), interaction between transient eddies and a diffluent flow (Shutts 1983; Colucci 1985, 2001; Trenberth 1986; Mullen 1987; Nakamura and Wallace 1993; Nakamura et al. 1997; Luo 2000, 2005; Altenhoff et al. 2008), instability/nonlinearity of low-frequency circulation (Swanson 2000; Cash and Lee 2000), selective absorption of vorticity anomalies (Yamazaki and Itoh 2009), and diabatic forcing from moist processes (Pfahl et al. 2015). Some of these theories are conceptual and difficult to verify with data directly, whereas others are more diagnostic and do not have predictive skill. Incomplete theoretical understanding hinders interpretation of blocking statistics and its response to climate perturbation in simulations and reanalysis products.

Recently Nakamura and Huang (2018, hereafter NH18) proposed a semiempirical theory for block formation based on the observed budget of local wave activity (LWA). LWA is the amplitude of Rossby wave measured by the meridional displacement of quasigeostrophic potential vorticity (PV) from zonal symmetry (Huang and Nakamura 2016, 2017). NH18 showed that major blocks that develop in the exit regions of the storm tracks in the Northern Hemisphere winter are associated, on average, with a converging along-stream flux of LWA, which is dominated by zonal advection (Figs. 4 and 5c of NH18). The convergence occurs when an increasing LWA decelerates westerly winds to the point that the advective LWA flux stops growing, which defines the “capacity” of the jet stream for the Rossby wave transmission. The dynamics captured by their observation and theory is akin to that of traffic flow on a highway. Just as traffic jams form when traffic capacity of the highway is exceeded, blocks manifest when/where the capacity of the jet stream is reached for the Rossby wave traffic. As will be explained more fully in the next section, stationary waves modulate the capacity to transmit transient waves by creating confluence and diffluence of the jet stream (analogous to the variation in the speed limit on a highway) and localize block formation.

NH18 conceptualize this mechanism with a simple 1D nonlinear partial differential equation (PDE) [their Eq. (4), also Eq. (1) in the next section]. The model takes the stationary wave activity, transient eddy forcing, and the background group velocity (jet speed plus intrinsic group velocity) as prescribed external parameters and predicts the evolution of LWA associated with transient waves. It is capable of reproducing the salient features of the average North Atlantic blocking in winter (cf. Figs. 5 and 6 of NH18). The threshold of block formation is expressed in terms of the parameters of the model [their Eq. (5), also Eq. (4) below]. NH18 hypothesize that climate change affects blocking statistics by modifying the threshold condition. While we do not know precisely how climate change will affect the threshold, the model allows for a controlled variation of “climate states” spanned by the aforementioned parameters. The economy of the model is suited for conducting large-ensemble experiments over a wide range of parameter space, which is the main goal of this study. It is hoped that the behavior of model’s statistics with respect to the threshold will give us a clue as to how blocking might respond to hypothetical climate perturbations. Given the simplicity of the model, we do not expect the result to provide a quantitatively accurate prediction of future blocking trends, and the model parameters considered herein are likely not independent of each other under the real climate change. Instead, our intention is to provide a theoretical basis for understanding the blocking behaviors for a broad range of climate conditions. The next section reviews the 1D model introduced by NH18 and describes the experimental design. Section 3 discusses the results, followed by a summary and discussion in the concluding section.

## 2. The model

*A*(≥0) measures the meridional displacement of PV from zonal symmetry,

**F**is the generalized (3D) Eliassen–Palm (E-P) flux density including the zonal advection of LWA by a hypothetical wave-free reference state flow

*U*

_{REF}, and

*A*. NH18 construct a column budget of LWA from Eq. (2) and evaluate it with reanalysis data in the exit regions of the jet stream, identifying dominant terms. They find that the convergence of the zonal advective flux dominates the tendency of LWA on synoptic time scales and thus lump together the other unimportant terms into forcing and damping of LWA. Furthermore, they partition

*A*(

*x*,

*t*) into stationary and transient wave components as

*A*

_{0}is defined as the minimum value of

*A*at

*x*. By subtracting the equation for

*A*

_{0}(

*x*) from that for

*A*, they arrive at Eq. (1) (see the supplementary materials of NH18).

*u*is the zonal wind and

*U*

_{REF}is a constant zonal wind of the wave-free reference state. The constant parameter

*α*measures the strength of wave-zonal flow interaction. Using reanalysis data NH18 determined

*α*≈ 0.55 in the exit regions of the storm tracks in the boreal winter.

^{1}The

*C*(

*x*) ≡

*U*

_{REF}+

*c*

_{g}− 2

*αA*

_{0}(

*x*) is the background group velocity in the reference state,

*U*

_{REF}+

*c*

_{g}, modulated by the stationary wave

*A*

_{0}(

*x*). Half of the modulation reflects the deceleration of the advecting zonal wind by the stationary wave, whereas the remaining half reflects direct interaction of transient waves with the stationary wave. Note that the modification of the meridional PV gradient by the waves is implicit in

*A*[generally, large

*A*is associated with a reduced meridional PV gradient; Eq. (18) of Huang and Nakamura 2016]. As a packet of eastward propagating Rossby waves encounters a diffluent region of the jet stream maintained by the stationary wave, where

*C*(

*x*) is small, the packet slows down and accumulates

*F*. However it also decelerates the zonal wind through Eq. (3), which acts to diminish the advective flux of

*F*

_{max}=

*C*

^{2}/4

*α*, at

*x*,

*t*) defines the saturation level of the transient LWA flux

*F*(NH18). As

*F*starts to decrease and causes a runaway accumulation of wave activity, provided that there is a continued supply of wave activity from upstream. This leads to a rapid increase in

*F*, which characterize block formation (Nakamura and Huang 2017; NH18). Thus Eq. (4) may be viewed as a threshold for blocking onset. It is clear that a small

*C*(

*x*) requires only a small

*F*and

*C*(

*x*) locally, which typically correspond to the diffluent regions of the jet stream.

The second term on the rhs of Eq. (1), *τ*. The last diffusion term is related to the correlation between PV and the zonal wind along the meridional path of displacement (see appendix A). This term also keeps the numerical solution smooth.

Equation (1) is highly idealized and not meant for accurate prediction of blocks in real weather. The model provides no direct connection to temperature or precipitation anomalies. Since the Rossby waves in the model only propagate zonally or are absorbed by blocks without being refracted meridionally, some details of blocking life cycle are likely misrepresented. Still Eq. (1) encapsulates canonical dynamics that produce persistent anomalies in the jet stream inferred from data, and it reproduces the salient features of blocking in the boreal winter when the threshold is reached (NH18). Nakamura and Huang (2017) also show that realistic wave breaking and blocking occur in a 2D model once the same threshold is reached, and that the zonal structure of the wave envelope is qualitatively similar to the 1D result (their Figs. 3 and 10).

Yet the true utility of Eq. (1) lies in its economy: it is suited for large ensemble runs in parameter sweep experiments and long-term calculations. We expect that the blocking threshold will shift when the parameters of the model are varied and affect the statistics of blocks. In the subsequent experiments we will vary transient eddy forcing *A*_{0}(*x*) and the background group velocity *U*_{REF} + *c*_{g} in extended integrations of Eq. (1), and examine how the statistics of blocking responds to these “climate variations.” We fix the other parameters as *τ* = 10 days, *α* = 0.55 and *κ* = 3.26 × 10^{5} m^{2} s^{−1}. The first two values are based on our previous observational analyses (Huang and Nakamura 2017; NH18), whereas the last value is consistent with Nakamura and Huang (2017). A periodic channel is assumed with a length of *L*_{x} = 2.8 × 10^{7} m. We have tested both finite-difference and spectral transform methods and obtained virtually identical results. The results shown in the next section are based on the spectral transform method with 1024 grid spacings. We use an exponential time differencing method with a fourth-order Runge–Kutta scheme (Cox and Matthews 2002; Kassam and Trefethen 2005) and a time increment of 432 s. We describe below the specific forms of the parameters to be varied. The list of parameters and the range of their values are found in Table 1.

List of parameters.

### a. Transient eddy forcing

*γ*controls the overall strength of forcing, and the variables with subscript

*n*are picked randomly at the beginning of each simulation (and kept fixed during the simulation) with the following rules:

- The variable
*N*= 26 and the zonal wavenumber of the component waves*k*_{n}is uniformly sampled from a set of integers between 1 and 20. - Frequency of the component waves
*ω*_{n}(s^{−1}) is uniformly sampled from [−2*π*, 2*π*] × 5.787 × 10^{−7}. - Phase of the component waves
*ϕ*_{n}is uniformly sampled from [0, 2*π*]. - Amplitude of the component waves
*w*_{n}is uniformly sampled from [0, 3.7].

*S*

_{0}specified in NH18 [their Eq. (S14)]. The above choice of parameters is determined by a trial-and-error method with the following guidelines: (i) forcing should be spectrally broad but limited to a range of wavenumbers and frequencies representative of synoptic events; (ii) within this spectral range, combination of the parameters should be randomized so that after many realizations a broad range of the forcing spectra is sampled. The purpose is to ensure that a robust blocking statistics emerges from a stochastic forcing, not from a narrowly prescribed frequency and length scale. Figure 1a shows a typical distribution of

*γ*. For a given

*γ*we repeat the simulation 240 times, each time randomizing the above phase parameters. In this sense the obtained ensemble of simulations is “pseudostochastic,” even though Eq. (1) is deterministic.

### b. Stationary wave

*k*and noise-induced amplitude modulation

*μ*are the additional degrees of freedom in the stationary wave. The latter is meant to mimic fluctuations in the stationary wave through boundary forcing. Similar to

*μ*(

*x*,

*t*) is given in terms of superposition of interfering waves, where

*M*= 21 and*k*_{m}is uniformly sampled from a set of integers between 1 and 10;*ω*_{m}is uniformly sampled from [−2*π*, 2*π*] × 2.894 × 10^{−7}(s^{−1});*ϕ*_{m}is uniformly sampled from [0, 2*π*];*w*_{m}is uniformly sampled from [0, 1].

*m*are randomized at the beginning of each simulation. The addition of the noise modifies

*C*(

*x*) in Eq. (1) to

*C*(

*x*,

*t*) =

*U*

_{REF}+

*c*

_{g}− 2

*αA*

_{0}(

*x*,

*t*). Although the time dependence of

*A*

_{0}may seem to obscure the distinction between stationary and transient waves, it represents a fundamentally distinct forcing process (boundary forcing as opposed to the internal dynamics) and operates at a slower time scale than the transient eddy forcing (

*ω*

_{m}<

*ω*

_{n}). We control the amplitude of the stationary wave through Λ and

*ε*.

### c. Jet speed

The sum *U*_{REF} + *c*_{g} denotes the average group velocity of Rossby waves in a hypothetical, wave-free reference state, including advection by the zonal wind *U*_{REF}. In a baroclinic atmosphere *U*_{REF}(*y*, *z*) would be inverted from the reference-state PV, *Q*_{REF}(*y*, *z*), which in turn would be obtained by zonalizing a wavy, instantaneous PV field through area-preserving map (Nakamura and Zhu 2010). In the absence of nonconservative processes *U*_{REF} would be invariant in time; in reality it varies slowly in response to diabatic heating, mixing, frictional damping, etc., and shows large seasonal variation (Nakamura and Solomon 2010, 2011; Methven and Berrisford 2015). Since we are concerned with blocking statistics under a characteristic flow condition, we prescribe *U*_{REF} + *c*_{g} as a constant for each simulation. Figure 4 of NH18 (orange diamonds) indicates that the background group velocity is nearly constant over a wide range of LWA. Its variation primarily represents changes in the jet speed *U*_{REF} in response to seasonal to decadal climate forcing. The change in the background PV gradient can also affect *c*_{g}, but in the storm-track regions the magnitude of *c*_{g} is generally much smaller than *U*_{REF} (not shown) and so the latter effect is thought to be minor. For this reason, we will subsequently refer to *U*_{REF} + *c*_{g} ≡ *U*_{J} as “jet speed.”

Figure 1b shows a typical structure of *C*(*x*, *t*) for *U*_{J} = 60 m s^{−1}, Λ = 10 m s^{−1}, *k* = 2, and *ε* = 0.5. The predominant stationary wavenumber 2 creates two minima in *C*(*x*, *t*), while the effect of noise is minimal with this parameter choice.

### d. Nondimensional parameters

*x*and

*t*by

*L*

_{x}and

*C*with

*U*

_{J}, the following five nondimensional parameters emerge:

*a*

_{1}and

*a*

_{2}measure the magnitude and wavenumber of stationary wave activity;

*a*

_{3}is the magnitude of transient eddy forcing;

*a*

_{4}and

*a*

_{5}quantify the strengths of damping and diffusion. We fix

*L*

_{x},

*α*,

*S*

_{0},

*τ*, and

*κ*in all of the subsequent experiments (and

*ε*and

*k*for most experiments), but

*U*

_{J}affects all but one of the parameters when we vary the jet speed. For an intuitive interpretation, in what follows we characterize our results with the three (dimensional) parameters, 2

*α*Λ,

*γ*, and

*U*

_{J}, instead of the above nondimensional parameters. Again see Table 1 for the list of parameters.

## 3. Results

The above model is used to simulate block formation in extended runs. For each run, we choose a combination of 2*α*Λ, *γ*, and *U*_{J}, and randomize the phase parameters of transient eddy forcing and stationary wave noise. The model is run for 270 days without the transient eddy forcing or stationary wave noise to reach a steady state, at which point we switch on the transient forcing and noise. Subsequently the model is run for another 180 days and we identify blocking events by the detection algorithm described below. We repeat the transient part of experiment 240 times for the same combination of 2*α*Λ, *γ*, and *U*_{J}, each time randomizing the phase parameters. We write out the snapshot of LWA to the disk at every 50 time steps (36 min) for postprocessing.

### a. Illustrative results

Figure 2 illustrates solutions for two representative flow regimes in Hovmöller diagrams. The first three panels in the top row show the total LWA, *u* ≡ 40 − *αA* (Figs. 2b), and the transient LWA flux *F* (Figs. 2c) for a weak stationary wave forcing (2*α*Λ = 0.8 m s^{−1}, *γ* = 3, *U*_{J} = 40 m s^{−1}, *k* = 2, and *ε* = 0.5). In this case, transient eddy forcing generates a series of wave packets that migrate downstream, characterized by diagonal stripes in all variables. Due to the imposed damping (*τ* = 10 days) the packets have finite lengths, and the LWA flux (Fig. 2c) is approximately proportional to LWA (Fig. 2a). Occasionally strong forcing events cause the threshold [Eq. (4)] to be exceeded, indicated by black contours in Fig. 2a. The majority of these supercritical episodes are short-lived and minor, and they occur throughout the channel, as they are a direct response to the forcing events. In this particular realization, a few significant episodes occur in the upstream of the weak stationary LWA ridges with substantial accumulation of LWA (Fig. 2a), deceleration of zonal wind (Fig. 2b) and precipitous drop in the LWA flux (Fig. 2c), but these episodes are generally infrequent in this parameter setting.

The bottom row (Figs. 2e–g) shows a case with stronger stationary wave and faster jet (2*α*Λ = 11 m s^{−1}, *γ* = 2, *U*_{J} = 60 m s^{−1}, *k* = 2, and *ε* = 0.5). The model climate in this case is dominated by persistent anomalies in all variables. Although forcing events are patchy and spread over the entire domain (Fig. 1a), regions that exceed the threshold form and persist primarily in the vicinity, and slightly upstream, of the stationary LWA ridges where the zonal wind is perpetually weak. Large-amplitude anomalies coincide with these regions, and they have sharp, shock-like upstream edges (Nakamura and Huang 2017), at which the downstream LWA flux is disrupted abruptly (Figs. 2e–g). Anomalies grow by expanding the upstream edges westward by absorbing the incident LWA flux (Figs. 2e,f) before they die out. That this case produces persistent anomalies despite the much weaker transient eddy forcing than the previous case (*a*_{3} = 1.3 as opposed to 9.1) highlights the importance of stationary wave and the associated streamwise variation in the wind speed in producing and localizing the persistent anomalies. We will present a more comprehensive parameter sweep shortly.

### b. Detection method and metrics

To construct statistics of blocking, one needs a method to identify and count the blocking episodes. In designing a detection method, we considered two criteria: (i) a blocking event should be roughly bounded by the threshold condition [Eq. (4)] but also reflect some maturity; (ii) instead of defining a block by an arbitrary minimum duration, we wish to obtain a distribution of frequency and persistence of episodes. In the end, the following method has been adopted, along with three metrics of blocks: (i) frequency, (ii) prevalence, and (iii) persistence. We first identify the grids in the longitude–time domain of the experiment (*L*_{x} × 180 days) at which the magnitude of ∂*u*/∂*x* (or equivalently ∂*A*/∂*x*) exceeds five standard deviations. They typically coincide with the upstream edges of blocking episodes.^{2} Next we apply a binary mask, namely, assign a value of 1 to these grids and 0 everywhere else. We will then run a counting algorithm outlined in appendix B, which identifies a single grid at the onset of each blocking episode. We count these grids in the domain to determine the *frequency* of blocks. To measure the overall *prevalence* of blocks, we count the total number of the grids with the value of 1 in the domain (blocking pixels). We then evaluate the average *persistence* of individual blocks by dividing prevalence by the number of blocks in the domain.

Figures 2d and 2h show the blocking pixels (wiggly strings) and the onset grids (stars) for the two experiments with weak and strong stationary wave forcing. Even though the former identifies many more blocks (18 vs 8), most of them have short persistence.

### c. Parameter sweep

Figure 3 summarizes the result of parameter sweep experiments. Each column describes the response of blocking statistics to the variation of one parameter, using the three metrics introduced above. In the left column, we vary the stationary wave amplitude 2*α*Λ with the other parameters fixed at default values (see the figure caption). When the stationary wave is weak, blocks are either rare or short-lived, so the prevalence is small. Both prevalence and frequency increase sharply as the stationary wave amplitude is increased beyond 2*α*Λ ~ 10 m s^{−1} (Figs. 3a,b), as the threshold [Eq. (4)] is fulfilled more frequently. Although the variation of prevalence is monotonic, frequency decreases some as the stationary wave amplitude is raised further (Fig. 3b). This is because blocks merge and become more persistent, as revealed in Fig. 3c. Frequency also exhibits greater uncertainty than prevalence partly due to its discrete nature (mostly single-digit integers). The uncertainty in frequency also causes uncertainty in persistence.

In the second column, we increase transient eddy forcing eightfold. In response, prevalence increases monotonically by a factor of 2 or so (Fig. 3d). Frequency averages around 6 (180 days)^{−1} when forcing is weak, and it increases slightly with forcing. For stronger forcing, however, frequency decreases significantly despite the continued increase in prevalence (Fig. 3e). This is due to a marked increase in persistence, by a factor of 3 over the range of forcing examined (Fig. 3f). Figure 4 is similar to the bottom row of Fig. 2 but with a stronger eddy forcing. In this case blocking becomes nearly perpetual in the vicinity of the two stationary ridges in LWA (Fig. 4d), and the zonal wind and flux often reverse to westward (Figs. 4b,c).

The response of blocking to the jet speed variation is roughly the opposite of the response to the stationary wave amplitude (Fig. 3, right column): prevalence and frequency decrease sharply beyond *U*_{J} ≈ 60 m s^{−1} (Figs. 3g,h). This is understandable since increasing the jet speed has the same effect as decreasing the stationary wave amplitude in *a*_{1} [Eq. (9)]. However, in the limit of slow jet speed, prevalence is significantly higher than in the limit of large stationary wave amplitude (Fig. 3g vs Fig. 3a). This is because the decreasing jet speed effectively increases transient eddy forcing *a*_{3} relative to damping and diffusion terms *a*_{4} and *a*_{5} [Eq. (9)]. Persistence also decreases, albeit more slowly, until *U*_{J} ≈ 75 m s^{−1}, beyond which blocks virtually disappear (Fig. 3i).

Figure 5 summarizes the blocking statistics as functions of stationary wave amplitude and jet speed. All three metrics show clear transition from a nearly block-free state to a block-dominant state. When the jet is slow, this transition occurs at small stationary wave amplitude. As the jet speed increases, larger stationary wave amplitude is required for the transition. A slight exception to this rule is frequency at small stationary wave amplitude, which shows weak secondary maximum along the vertical axis (Fig. 5b). This is because when the stationary wave is weak and eddy forcing is reasonably strong, blocks with short persistence arising directly from forcing prevail (see Fig. 2d).

We have also tested the parameter dependence in the context of an initial-value problem. Figure 6 shows the result of a 145-yr run, in which the jet speed *U*_{J} is decreased gradually from 70 m s^{−1} at a rate of 0.17 m s^{−1} yr^{−1}, while other parameters are fixed (2*α*Λ = 11 m s^{−1}, *γ* = 2, *k* = 2, *ε* = 0.5). In this case we have randomized the phase of the forcing at certain intervals to avoid inadvertent periodicity. Each panel shows the total LWA

### d. Dependence on other parameters

In addition to the three parameters, we have also varied the wavenumber *k* [Eq. (7)] and the noise level *ε* [Eq. (8)] of the stationary wave. When the wavenumber of stationary wave is increased from 1 to 10, prevalence dips initially (1≤ *k* ≤ 3) and then turns upward (Fig. 7a). However, the overall variation of prevalence is modest (the maximum at *k* = 10 is only 33% above the minimum at *k* = 3). Frequency, on the other hand, shows a nearly linear, threefold increase over 1≤ *k* ≤ 10 (Fig. 7b). This is because the increasing wavenumber adds more stationary ridges in LWA that are traffic bottlenecks and conducive to blocking. However, since prevalence does not change much, blocks become more numerous but shorter. Persistence decreases by about 50% over 1≤ *k* ≤ 4 although the subsequent reduction is very modest (Fig. 7c). Overall, the wavenumber of the stationary wave affects the frequency (and locations) of blocking the most.

When we increase the noise level of the stationary wave amplitude *ε*, both prevalence and frequency increase rapidly at first and gradually level off as *ε* increases further. Persistence, on the other hand, remains more or less steady (Figs. 7d–f). Comparisons with Figs. 3a, 3b, and 5 suggest that the enhanced *ε* effectively increases the amplitude of stationary wave and accelerates the transition from the block-free state to the block-dominant state, greatly affecting the frequency and prevalence in the transition zone.

## 4. Summary and discussion

Current climate models do not exhibit high confidence in the projection of blocking frequency under a changing climate (Woollings et al. 2018). Even discounting the systematic biases and other shortcomings of the model simulations, building reliable statistics of blocking events, let alone evaluating the nonstationary aspect of it, is inherently resource intensive due to the intermittent nature of blocks. Given this, there is a virtue in studying blocking statistics using a simple model that grants computational economy and theoretical interpretations.

We have constructed and analyzed a large ensemble of 180-day simulations in the parameter space of the “traffic jam” model of blocking [NH18; Eq. (1)]. In this simple 1D model, transient waves are generated by pseudostochastic eddy forcing and allowed to interact with the westerly wind and stationary wave. Although the statistics of transient eddy forcing is homogeneous, blocks form by selectively collecting the wave activity flux from significant forcing events in the upstream of the stationary LWA ridge. The majority of blocks in this model therefore form in the vicinity (or slightly upstream) of the stationary LWA ridges, where the westerly wind is always weak. This matches the climatological locations of the major blocks in the Northern Hemisphere, which center around the exit regions of the storm tracks (Woollings et al. 2018). The boundaries of blocks generally coincide well with the threshold condition [Eq. (4)]. Therefore, the modulations of the threshold condition due to changes in the parameters (climate variations) affect the blocking statistics.

Blocking statistics in this model proves particularly sensitive to the stationary wave amplitude and the jet speed. For a given transient eddy forcing, the model’s climate shifts quickly from a block-free state to a block-dominant state as the stationary wave amplitude is increased and/or the jet speed is reduced. Proximity to the blocking threshold is determined by the ratio of LWA to the jet speed. As the stationary wave LWA increases, less additional transient wave LWA is required to fulfill the threshold. As a result, a greater stationary wave is more conducive to block formation for the given jet speed and transient eddy forcing. Similarly, a slower jet makes the flow closer to the threshold and thus more conducive to block formation for the given stationary wave and transient eddy forcing.

An increasing transient eddy forcing also promotes blocking, and its main effect in the presence of stationary waves is to make blocking fewer and more persistent (and eventually perpetual). Causes of transient eddy forcing include baroclinic cyclogenesis in the upstream (Colucci 1985), diabatic heating associated with moist processes (Pfahl et al. 2015), and merger of storms (Riboldi et al. 2019). Given that we observe blocks frequently but not perpetually in the real atmosphere, we speculate that the present climate lies close to the transition between block-free and block-dominant states. Furthermore, we envision that the seasonal variation (strong stationary waves and a fast jet in winter; weak stationary waves and a slower jet in summer) moves the state of the atmosphere along, but not across, the regime boundary, so blocking is observed year-round. Since the boundary is sharp with respect to the stationary wave amplitude and jet speed, blocking in climate models is likely sensitive to these quantities if the present climate indeed lies in the vicinity of the regime boundary. Many CMIP5 models underestimate the frequency of Atlantic blocking, and even though they collectively predict a weak decreasing trend in future blocking frequency, the confidence level of the projections is low (Masato et al. 2013; Woollings et al. 2018). A positive bias in the jet speed and/or a negative bias in the stationary wave amplitude are the prime candidates that suppress the Atlantic blocks in these models. If these biases in large-scale circulation patterns can be corrected, blocking statistics in the climate models may improve significantly (Scaife et al. 2011). However, the cause of the biases is nontrivial: for example, poor resolution of orography and moist convection affects the stationary wave amplitude, and inadequate representation of sea surface temperature affects the overall mean states of the storm tracks (Berckmans et al. 2013). Because of the intricate interplay among the internal processes, correcting for the bias requires a careful and holistic examination of the model dynamics. Understanding how flow parameters influence various aspects of blocking is just the starting point; it provides a theoretical framework for addressing the model biases, as well as interpreting the observed trends in blocking statistics.

Given the highly idealized nature of Eq. (1), some details of the blocking dynamics it represents merit further scrutiny. In particular, the demise phase of blocking likely involves meridional transmission of Rossby wave packets, and expressing it as linear damping with a constant damping time scale is a gross oversimplification. How a more elaborate representation of wave activity fluxes affects the persistence of blocks is a topic worthy of future study.

In this work we focused on the local interaction between transient waves and zonal flow as a mechanism of block formation, wherein the stationary wave amplitude was prescribed. A complementary mechanism of block formation is resonance between the boundary forcing and stationary waves (Charney and DeVore 1979; Tung and Lindzen 1979; Brunet 1994; Petoukhov et al. 2013). While resonance is discussed primarily in the context of amplification of the stationary wave, our study suggests that an enhanced stationary wave can in turn affect blocking through interaction with transient waves, by modulating the threshold. This is implied in section 3, where perturbation to the stationary wave resulted in a significant change in the blocking statistics. Relative importance of the traffic jam and resonance dynamics will be investigated in subsequent works.

The main results of this paper emerged from a group project during *Rossbypalooza*, a student-led summer school at the University of Chicago in June 2018, with the theme of “Understanding climate through simple models.” The authors thank the participants of the summer school for their valuable feedback. Constructive criticisms of the two anonymous reviewers greatly improved the quality of the manuscript. The work is supported by NSF Grants AGS1563307 and AGS1810964.

# APPENDIX A

## The Diffusive Term in Eq. (1)

*y*′ relative to the wave-free reference state at

*y*, and

*y*+

*η*(

*x*,

*y*,

*t*) is the instantaneous meridional location of the PV contour

*q*=

*Q*

_{REF}at

*x*[i.e.

*q*

_{e}(

*x*,

*y*+

*η*,

*t*) = 0]. Now define the average along the displacement path and departure from it as

*A*= −

*η*⟨

*q*

_{e}⟩ and Eq. (A1) may be rewritten as

*ξ*

^{†}(

*x*,

*y*+

*y*′,

*t*), such that

*κ*> 0, giving rise to the diffusive term in Eq. (1). In deriving Eq. (1), we also assumed ⟨

*u*

_{e}⟩ ≈ −

*αA*.

# APPENDIX B

## Counting of Discrete Patches

We implement a simple algorithm inspired by the classical computation problem, the Game of Life (Gardner 1970), to count discrete patches in a 2D domain. We assume that the 2D field is binary, that is, every grid point is assigned a value of either 0 or 1. We will count the number of contiguous areas represented by the grid points with 1. To do this, we scan the entire domain using a five-point stencil [(*i*, *j*), (*i* + 1, *j*), (*i* − 1, *j*), ( *i*, *j* + 1), ( *i*, *j* − 1)], where *i* and *j* are the indices of grids in the domain. In our case, the scan is performed along the row from top down, starting from the highest *j* to the lowest *j* (i.e., backward in time, forward in longitude). As we move the stencil across the domain, if the grid (*i*, *j*) has a value of 1, we check the values at the neighboring four grids. If any other grid has a value of 1, the value at (*i*, *j*) is reset to 0. This means that when the stencil reaches the last grid in a patch, only one grid is left from that patch with a value of 1. Once the stencil has reached the end of the domain, we sum the value of the entire grid. Since we now have one grid with a value of 1 per patch and 0 everywhere else, this sum is the number of patches. See the animation in the supplemental material.

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^{1}

In the WKB limit of a plane wave in barotropic flow where Kelvin’s circulation is conserved over one wavelength of the PV contour, Eq. (3) reduces to the local nonacceleration relation with *α* = 1 (Huang and Nakamura 2016; Nakamura and Huang 2017).