## 1. Introduction

Atmospheric blocking represents a disruption of eastward migration of synoptic eddies and often causes persistent weather anomalies in the midlatitudes. Distribution and frequency of blocking episodes, together with their trend in a changing climate, are under active investigation (Barriopedro et al. 2006; Croci-Maspoli et al. 2007; Tyrlis and Hoskins 2008; Masato et al. 2013; Barnes et al. 2014; Mokhov et al. 2014). Blocking itself is a low-frequency phenomenon that can persist for a week or longer, but the role of high-frequency (synoptic) eddies and their feedback on the low-frequency dynamics in the block formation has been noted for some time (Berggren et al. 1949; Green 1977; Shutts 1983 and references therein; Colucci 1985, 2001; Mullen 1987; Altenhoff et al. 2008). Nakamura et al. (1997) show that the feedback from synoptic eddies accounts for the majority of observed block formation over the North Pacific but less than half of blocking events over Europe. A number of theoretical studies based on barotropic dynamics demonstrate that local interaction between a preexisting stationary diffluent flow and incident transient eddies can amplify a blocklike quasi-stationary feature (Shutts 1983; Haines and Marshall 1987; Anderson 1995; Luo 2005). However, identifying a precise condition that initiates blocking formation remains elusive.

In this article we revisit equivalent barotropic dynamics of a potential vorticity (PV) front, in which a train of short edge waves enters the domain from upstream and interacts with a diffluent jet downstream. It will be shown that this idealized model predicts a clear transition from steady downstream propagation of waves to highly deformed, blocklike wave tracks, depending on the upstream wave activity flux and the geometry of the incipient flow. The model dynamics is similar to the one considered by Swanson et al. (1997) except for two aspects. First, transient waves are fed continuously from upstream rather than as an isolated wave packet, and second, the layer depth is finite and allowed to vary meridionally to mimic the isentropic structure of PV near the extratropical tropopause. The finite Rossby radius only affects the basic state and the low-frequency response of the flow, since the transient wave forcing is in the shortwave (barotropic) limit and insensitive to the Rossby radius.

Another departure from Swanson et al. (1997) is that, in addition to the conservation of wave action (Bretherton and Garrett 1968), we will exploit the conservation of local wave activity (LWA) and associated local nonacceleration relation (Huang and Nakamura 2016, hereafter HN16) to predict the threshold behavior of wave envelope in a 1D quasilinear theory. As we will see below, given a longitudinally varying initial zonal flow and an upstream influx of wave activity, there is a threshold LWA beyond which no steady wave envelope exists. This allows one to predict the location of the onset of wave breaking. The line of argument is akin to Fyfe and Held (1990) for meridionally propagating Rossby waves and Wang and Fyfe (2000) for vertically propagating edge waves. We will then demonstrate numerically, both in 1D and fully nonlinear 2D calculations, that the wave amplitude undergoes a significant transformation once this threshold is reached. When the PV jump has significant contribution from the variation of layer thickness, the resulting 2D flow captures salient features of an atmospheric blocking in a manner similar to Shutts (1983).

The next section outlines the 1D theory. Section 3 describes the 2D model and the results of numerical experiments. Relevance to the observed atmospheric blocking will be discussed in the concluding section.

## 2. One-dimensional quasilinear theory for a slowly varying flow

*f*plane

*g*and average layer thickness

*H*;

*y*= 0,

*x*. A specific form of

*U*varies only slowly in

*x*, (2) admits a steady WKB solution in the form

*ε*≪ 1) is the zonal coordinate of slow variation and

*ω*is independent of

*X*since

*ω*and

*k*are related through a phase function

*ω*is independent of time since the basic state is steady (Bretherton and Garrett 1968). The corresponding dispersion relation is (e.g., Zhu and Nakamura 2010)

*kL*

_{D}≫ 1 (i.e.,

*κ*≈

*k*). While this choice makes the theory mathematically tractable, we argue in section 4 that it is not the critical aspect of the underlying physics. In this limit (8)–(10) are approximated by the following barotropic formula:

*ω*is independent of

*X*for a steady wave, (12) gives

*U*decreases. Meanwhile, from (11) and (12),

*X*even when

*U*varies with

*X*[Swanson et al. (1997), their appendix C]. Note that (16) and (17) hold only in the barotropic (shortwave) limit, whereas (14) and (15) apply more generally.

### a. Local wave activity and wave–zonal flow interaction

*u*that excludes fast oscillation associated with waves. In the presence of finite-amplitude waves it is

*U*. To understand the change from

*U*to

*c*is close to

*U*with

*ω*is constant in a steady state, this leads to

*X*:

*ω*and

*X*. It turns out that the plus root in (28) is unstable and thus unrealizable: if

*U*

^{2}/4) that can sustain a steady state at that longitude. At the threshold one observes that the zonal flow is decelerated to one-half of the original value; that is,

### b. Transient behavior

*ω*is no longer strictly constant, we use a fixed value of

*ω*(i.e., the forcing frequency) everywhere. The associated error is small as long as

*U*is specified as

*x** = 2

*π*) to let the waves exit the domain.

The domain is discretized into 4097 grids and a standard finite difference method is used. We also add a small second-order diffusion term to (30) for numerical stability. [The diffusion coefficient is

Figure 2 depicts the numerical solutions for *t** = 3.3, 6.6, 9.9, 13.2). With this choice of *t** = 13.2). The amplitude and wavenumber of the wave train are modulated according to (16): they are both greatest in the region where *t** = 13.2) is in excellent agreement with the theoretically predicted asymptotic zonal flow (green).

Longitudinal structure of a 1D transient wave train traversing a zonally varying flow at different instances. Moderate wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Longitudinal structure of a 1D transient wave train traversing a zonally varying flow at different instances. Moderate wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Longitudinal structure of a 1D transient wave train traversing a zonally varying flow at different instances. Moderate wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Figure 3 shows an analogous result for *t** = 3.3), but once the zonal flow approaches the threshold value near the center of the domain (i.e., the break in the green curve), the wave envelope begins to develop discontinuity (*t** = 6.6). The discontinuity continues to grow and migrates upstream (*t** = 9.9, 13.2). The LWA flux also becomes discontinuous at the same location, the downstream value being smaller than the upstream value of

As in Fig. 2, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 2, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 2, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

*X*, but a shock can form in a broad class of

Location of the shock in Fig. 3 as a function of time. Black indicates direct observation. The location of the shock is determined by the *n* is chosen to account for the finite width of the shock due to numerical diffusion. We use *n* = 4 for the black curve. The other curves are the predictions based on the time integration of the last expression of (37) from *t** = 7.5. The computed Rankine–Hugoniot speed does depend on the choice of *n*, albeit not too sensitively. Green: *n* = 3. Red: *n* = 4. Blue: *n* = 5. Gray: *n* = 6. In the present case *n* = 4 results in a best agreement with the observation. Note *n* = 3 eventually diverges from the observation, as it underestimates *t** = 11 and 13.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Location of the shock in Fig. 3 as a function of time. Black indicates direct observation. The location of the shock is determined by the *n* is chosen to account for the finite width of the shock due to numerical diffusion. We use *n* = 4 for the black curve. The other curves are the predictions based on the time integration of the last expression of (37) from *t** = 7.5. The computed Rankine–Hugoniot speed does depend on the choice of *n*, albeit not too sensitively. Green: *n* = 3. Red: *n* = 4. Blue: *n* = 5. Gray: *n* = 6. In the present case *n* = 4 results in a best agreement with the observation. Note *n* = 3 eventually diverges from the observation, as it underestimates *t** = 11 and 13.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Location of the shock in Fig. 3 as a function of time. Black indicates direct observation. The location of the shock is determined by the *n* is chosen to account for the finite width of the shock due to numerical diffusion. We use *n* = 4 for the black curve. The other curves are the predictions based on the time integration of the last expression of (37) from *t** = 7.5. The computed Rankine–Hugoniot speed does depend on the choice of *n*, albeit not too sensitively. Green: *n* = 3. Red: *n* = 4. Blue: *n* = 5. Gray: *n* = 6. In the present case *n* = 4 results in a best agreement with the observation. Note *n* = 3 eventually diverges from the observation, as it underestimates *t** = 11 and 13.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

## 3. Two-dimensional nonlinear calculations

The preceding analysis shows that 1D quasi-linear edge waves develop a migratory shock in the envelope once the zonal flow is decelerated to one-half of the initial value. Since the discontinuity violates the premise of slow variation, the WKB approximation formally breaks down at this point. Similar loss of steadiness in wave activity was previously associated with the onset of wave breaking (Fyfe and Held 1990; Wang and Fyfe 2000). Since the shock formation may be viewed as a botched attempt by the wave envelope to overturn (i.e., to become multivalued), it indeed suggests an onset of wave breaking, or at least significant deformation of the wave envelope, in the full 2D problem. For a more accurate description of the flow evolution past the threshold, we shall perform direct numerical simulations in 2D. While the contour advection algorithm (e.g., Dritschel and Ambaum 1997) would be a natural extension to the above linear analysis, we adopt a finite-difference model on a rectangular domain instead, in which PV front is allowed to have a narrow but finite width similar to Harvey et al. (2016). The use of the finite-difference model is partly motivated by the ease with which to implement the open boundary condition. To describe the eddy–zonal flow interaction in as clean a setup as possible, transient waves are forced upstream and allowed to migrate downstream and exit the domain without reentering, just as in the 1D problem discussed earlier.

*A*and

*B*are determined such that

*δ*, the PV jump has contributions from both the jet and from the change in the layer thickness:

December–February climatology of the Lagrangian-mean state on the 340-K isentropic surface as functions of equivalent latitude *a* = 6378 km is the planetary radius. Note *C* is the absolute circulation evaluated as the area integral of absolute vorticity over the domain demarcated by the PV contour. The rate of planetary rotation *M* is the area integral of *p* and *θ* are pressure and potential temperature, respectively) over the same PV domain and *S* is the area of that domain. (c) Ertel PV computed as

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

December–February climatology of the Lagrangian-mean state on the 340-K isentropic surface as functions of equivalent latitude *a* = 6378 km is the planetary radius. Note *C* is the absolute circulation evaluated as the area integral of absolute vorticity over the domain demarcated by the PV contour. The rate of planetary rotation *M* is the area integral of *p* and *θ* are pressure and potential temperature, respectively) over the same PV domain and *S* is the area of that domain. (c) Ertel PV computed as

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

December–February climatology of the Lagrangian-mean state on the 340-K isentropic surface as functions of equivalent latitude *a* = 6378 km is the planetary radius. Note *C* is the absolute circulation evaluated as the area integral of absolute vorticity over the domain demarcated by the PV contour. The rate of planetary rotation *M* is the area integral of *p* and *θ* are pressure and potential temperature, respectively) over the same PV domain and *S* is the area of that domain. (c) Ertel PV computed as

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Seasonal climatology of PV variation across the tropopause on the 340-K isentropic surface. The second column describes the range of latitudes in which PV varies most. The variation of PV in each latitude band relative to the global range of 340-K PV is shown in the third column. The last column shows the percentage contribution of the thickness variation to the PV variation within each latitude band; 50% corresponds to *β** = 1 in (50). Based on the 1979–2014 climatology computed with ERA-Interim Dee et al. (2011).

*y*, but we will consider the dynamics only in a finite domain in which

*δ*, where

*J*is the 2D Jacobian operator. For a transient problem we solve the full PV equation

*ν*is hyperviscosity to diffuse PV at small scales.

*π*and 0.6

*π*, respectively. The zonal length of the domain is identical to the 1D case. We pad this rectangular domain with a sponge layer all around, extending the computational domain by 20% in each dimension (the width of the sponge layer is 10% of the physical domain at both ends). In the sponge layer a linear damping is applied to

### a. Weak forcing without thickness variation

First, we assume that there is no thickness variation

PV and streamfunction at different stages of the 2D numerical experiment with *t** = 3.3, 6.6, 9.9, 13.2. The domain shown excludes the sponge layer. (left)

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

PV and streamfunction at different stages of the 2D numerical experiment with *t** = 3.3, 6.6, 9.9, 13.2. The domain shown excludes the sponge layer. (left)

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

PV and streamfunction at different stages of the 2D numerical experiment with *t** = 3.3, 6.6, 9.9, 13.2. The domain shown excludes the sponge layer. (left)

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

Figure 7 shows the corresponding changes in the zonal flow (left) and the LWA flux (right) at the axis of the jet

(left) The zonal-flow profile and (right) the zonal advective LWA flux *x** is applied to all curves, primarily to remove phase dependence of wave activity. If (24) is strictly obeyed, the gray and dashed curves should coincide with each other. See text for details.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

(left) The zonal-flow profile and (right) the zonal advective LWA flux *x** is applied to all curves, primarily to remove phase dependence of wave activity. If (24) is strictly obeyed, the gray and dashed curves should coincide with each other. See text for details.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

(left) The zonal-flow profile and (right) the zonal advective LWA flux *x** is applied to all curves, primarily to remove phase dependence of wave activity. If (24) is strictly obeyed, the gray and dashed curves should coincide with each other. See text for details.

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

### b. Strong forcing without thickness variation

The above case produces a steady wave envelope reminiscent of the quasilinear WKB solution. Now we increase the forcing to ^{−1}) owing to the smoothing of the PV front, and accordingly (29) is first violated farther downstream (*x** < 3.2) the zonal LWA flux decreases rapidly downstream, qualitatively similar to Fig. 3 (Fig. 9;

As in Fig. 6, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 6, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 6, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but with a stronger wave forcing

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

The separatrices of

### c. Strong forcing with thickness variation

In the preceding case the transient waves decelerate the zonal flow to the point that a significant transformation occurs in the wave envelope, eventually leading to a new 2D steady state. The longitudinal location at which this transformation occurs is largely consistent with the prediction by the 1D theory. However, despite the enormous deformation of the PV contours, the corresponding change in streamfunction is modest. The zonal flow does decrease below the threshold but it remains positive at

Now we introduce a cross-stream thickness variation by letting *t** = 4–8). The flow slows down at *t** = 12–16). At this point the wave train is split into two tracks at the stagnation point: the negative anomalies move northward to wrap around the anticyclone and the positive anomalies move southward around the cyclone. The processes observed here are consistent with the barotropic straining mechanism described by Shutts (1983). The PV filaments are gradually entrained into the cyclone–anticyclone gyres from downstream and gradually mixed inside (Fig. 10;

As in Fig. 6, but for *q** = 2.5 and −2.5, respectively, and the drawn PV contours are

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 6, but for *q** = 2.5 and −2.5, respectively, and the drawn PV contours are

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 6, but for *q** = 2.5 and −2.5, respectively, and the drawn PV contours are

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 7, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

By *t** = 16 the incident transient waves are almost completely blocked. There is a precipitous drop (and reversal) of the zonal LWA flux from *x** ≈ 12 to 2.7, yet virtually no flux leaks out downstream of *t** = 16). The incident LWA flux is all absorbed by the block, which continues to expand meridionally and at the same time slowly migrate upstream (Fig. 10; *t** = 16). Similar upstream migration of a block due to transient wave forcing has been reported by Luo (2005). Unlike the previous cases the separatrices of streamfunction themselves undergo substantial reconfiguration and therefore their initial geometry fails to be a predictor of the subsequent behavior of the wave envelope. Eventually a quasi-steady state emerges when the incident LWA and energy are damped in the sponge layer. Some LWA is lost internally owing to efficient PV mixing inside the block, which is why the gray and dashed curves in the left column of Fig. 11 show appreciable difference after *t** = 12.

The role of the layer thickness variation in the block formation is qualitatively understood as follows. The meridional displacement of the PV contour induces a net (i.e., local phase average) PV anomaly whose sign is opposite across *y* = 0. Close to the jet axis, the net PV anomaly is about

### d. Parameter sweep for and

We have repeated the experiments for a range of forcing amplitude *a* above.) The plus sign denotes partial blocking, in which the zonal flow does drop below one-half of the initial value somewhere along *b* above). The letter B denotes blocking, in which there is a reversal of the zonal wind along *c*).

Model behavior as a function of

## 4. Summary and discussion

We have explored the role of interaction between a high-frequency transient wave train and a diffluent zonal flow in the low-frequency dynamics of an equivalent barotropic PV front. Central to the dynamics are (i) a positive feedback between the accumulation of LWA and the local deceleration of the zonal flow and (ii) the cross-stream variation of the layer thickness. The conservation of LWA for slowly modulated 1D quasilinear edge waves predicts that a steady WKB solution ceases to exist once the local zonal flow is decelerated below one-half of its original value [i.e., the violation of (29); see related discussions by Fyfe and Held (1990) and Wang and Fyfe (2000)]. The location at which this occurs may be predicted from the upstream LWA flux and the profile of the initial zonal flow downstream. In the 1D transient problem, the wave envelope develops a shock once this threshold is reached. An analogy to the traffic flow problem (Lighthill and Whitham 1955; Richards 1956) was noted.

We have tested the theoretical prediction with 2D nonlinear experiments, in which the zonal variation of the background flow is assumed to occur on the scale of the Rossby radius. The experiments agree with the theory in that once the predicted threshold is exceeded the wave envelope undergoes a significant transformation. When the PV discontinuity is associated entirely with the vorticity profile, the result is a partial deflection of LWA analogous to Swanson et al. (1997). In this case the wave envelope reorganizes into a 2D steady state, in which the waves are partially deflected sideways and partially transmitted downstream. When the PV jump is augmented by a sufficiently large thickness variation, the meridional displacement of PV leads to a self-organization of an anticyclone–cyclone pair across the jet axis, in a manner similar to atmospheric blocking. The zonal flow is reversed and the incident wave train is split in two tracks at the stagnation point. The incoming flux of LWA is entrained into the block and partially dissipated by mixing, but very little flux escapes to the downstream side of the block. From the last column of Table 1, the required threshold of thickness variation for block formation (~50%) is generally met across the extratropical tropopause.

The present work provides a paradigm in which a relatively clean threshold exists for a transition from a westerly jet to a blocked state in response to high-frequency wave forcing—a counterpart to the block formation through instability/nonlinearity in low-frequency dynamics (e.g., Swanson 2000; Cash and Lee 2000). In our last experiment (Fig. 10) the block evolution is very slow compared to the frequency of the wave forcing, yet the mechanism of feedback from high-frequency to low-frequency dynamics is not trivial. The phase structure of PV anomalies cascades to small scales in a diffluent shear flow, so each individual PV filament does not carry much dynamical significance. However, the envelope of the PV contours grows by absorbing LWA from upstream and this affects the large-scale circulation. Of the processes that control the wave envelope in our model, the zonal advective flux of LWA is most influential but PV mixing also plays an appreciable role.

We have chosen our model setup so that the theory takes on a mathematically simplest form. Some of the assumptions invoked in the theory, including the shortwave (barotropic) and high-frequency limits, may seem restrictive. However our 2D numerical solutions show that as the wave enters a diffluent region its phase is compressed, suggesting that the shortwave assumption is indeed more appropriate where the wave interacts with the flow. While specific results like (29) do hinge on this assumption, we believe that the overall behavior of the model, such as the existence of wave breaking threshold, will remain unaffected in a more general setup insomuch as the dynamics is characterized by the along-stream propagation of transient waves. For example, the shock formation in the wave envelope only requires that the wave activity flux have a maximum with respect to wave activity. This is a consequence of nonlinearity in wave–mean flow interaction and is unlikely to be altered by the details of the parameter setting. As a test for the robustness of the result, we have run the 2D model with a much lower forcing frequency (*ω** = 4) with other parameters unchanged from Fig. 10. The result is shown in Fig. 12. Compared with Fig. 10, the incident wave has a much greater wavelength, but its phase still collapses at the stagnation point, guiding finite-sized eddies along split paths. The eventual formation of blocking remains qualitatively similar to Fig. 10, although the streamfunction is appreciably noisier.

As in Fig. 10, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 10, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

As in Fig. 10, but for

Citation: Journal of the Atmospheric Sciences 74, 7; 10.1175/JAS-D-17-0029.1

The present work concerns only the formation stage of blocking—our numerical experiments did not produce a well-defined life cycle of blocking. Even as a model of block formation, the extent to which this idealized study is applicable as a predictive tool remains to be seen, because in reality wave forcing and the response of the flow are not readily separable. For example, deep intrusion of tropical air with low PV into high latitudes occurs regularly from explosive cyclogenesis in the Northern Hemisphere winter storm tracks and it provides potent forcing for blocking (Colucci 1985). In Fig. 10 the emerging blocking high and cut-off low have comparable strengths owing to symmetry in the dynamics, but the symmetry may be broken by an asymmetric flow profile, Earth’s sphericity, or nonquasigeostrophic effects (Nakamura 1993). The behavior of our model is also distinct from those of the barotropic models on the *β* plane used in previous studies (e.g., Shutts 1983; Luo 2005), which produce blocks without the need for thickness variation. It appears that the piecewise-uniform PV model without thickness variation is prohibitive for block formation because the produced PV anomalies tend to be passive and incapable of self-organizing into large eddies.

Mutual reinforcement of the wave activity accumulation and the local deceleration of the zonal flow is corroborated by meteorological data. A recent work by Huang and Nakamura (2017) examines the regional budgets of column-averaged LWA over the Pacific and Atlantic storm tracks in the northern winter using the ERA-Interim products. Their analysis shows that covariance of the column-averaged LWA and the column-averaged zonal wind is largely negative and strongest in the jet exit (diffluent) regions (see Fig. 4 of their supporting information). Furthermore, they also show that on synoptic time scales the tendency of the column-averaged LWA is dominated by the convergence of the zonal LWA flux. This is consistent with (30). That the magnitude of the LWA–zonal flow covariation is greatest in the diffluent part of the zonal flow suggests that synoptic to intraseasonal variability of LWA occurs through the zonal flux variation in a manner described in this article. Future work will explore statistical connections between the upstream LWA flux and the frequency and magnitude of large LWA events in the diffluent regions.

## Acknowledgments

This work has been supported by NSF Grant AGS-1563307. The ERA-Interim dataset used in this study (Dee et al. 2011) was obtained from the ECMWF data server (at http://apps.ecmwf.int/datasets/data/interim-AU2 full-daily/) with the horizontal resolution of 1.5°. Helpful discussions with Malte Jansen are gratefully acknowledged. Critiques by three anonymous reviewers also contributed to significant improvements in the presentation.

## APPENDIX

### Conservation of Quasilinear Local Wave Activity and Zonal Momentum

#### a. Local wave activity in a barotropic PV front

*y*, and

*x*where the contour is displaced to a positive

*μ*represents phase shift, the phase average of

#### b. Local zonal flow in a barotropic PV front

*f*plane reads

*p*is dynamic pressure and

*ψ*through

*x*

*P*and

*x*, the corresponding meridional velocity

*V*is formally

*U*, but in our problem we choose a flow in which

*P*and

*y*derivative of

*υ*′ since

*x*derivative of

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