## 1. Introduction

*A*is the density of wave activity (negative angular pseudomomentum);

*t*is time;

**F**is the generalized E–P flux, which represents radiation stress of the wave and equals the group velocity times the wave activity density for a slowly modulated, small-amplitude wave;

*D*denotes nonconservative effects on wave activity; and

*α*, a measure of wave amplitude. For a small-amplitude, conservative wave, the right-hand side terms are negligible, and wave activity density changes only where there is nonzero E–P flux divergence. The E–P flux divergence in turn drives the angular momentum of the mean flow, thus acting as the agent of wave–mean flow interaction.

Nakamura and Zhu (2010, hereafter NZ10) extended (1) for finite-amplitude Rossby waves and balanced eddies by introducing finite-amplitude wave activity (FAWA) based on the meridional displacement of quasigeostrophic potential vorticity (PV) from zonal symmetry. The formalism eliminates the cubic term from the right-hand side of (1) and extends the nonacceleration theorem (Charney and Drazin 1961) for an arbitrary eddy amplitude. This allows one to quantify the amount of the mean-flow modification by the eddy (Nakamura and Solomon 2010, 2011).

Despite its amenability to data, FAWA is a zonally averaged quantity and is incapable of distinguishing longitudinally isolated events, such as atmospheric blocking. In this article, we shall address this shortcoming by introducing local finite-amplitude wave activity (LWA). In essence, LWA quantifies longitude-by-longitude contributions to FAWA and, as such, recovers FAWA upon zonal averaging. As a first step into this topic, the present article concerns primarily the conservative dynamics of local eddy–mean flow interaction. Explicit representation of nonconservative dynamics (such as local diffusive flux of PV) will be deferred to a subsequent work. However, when observed data deviates from the theory, it may be readily interpreted as an indication of nonconservative effects. The material is organized as follows. Section 2 lays out the theory. Section 3 demonstrates the utility of LWA using idealized simulations with a barotropic vorticity equation on a sphere. We will compare LWA with one of the existing local metrics of finite-amplitude wave activity: impulse-Casimir wave activity (Killworth and McIntyre 1985; McIntyre and Shepherd 1987; Haynes 1988). As an application of LWA, a blocking episode that steered Superstorm Sandy to the U.S. East Coast in 2012 will be studied in section 4. Discussion and concluding remarks will follow in section 5.

## 2. Theory

### a. Finite-amplitude wave activity

*y*, pressure pseudoheight

*z*, and time

*t*in terms of surface integrals of quasigeostrophic PV over two domains of equal area:

*z*surface,

*Q*, whereas

*y*. The requirement that the areas of

*Q*and

*y*for a given

*z*. The latitude

*Q*is termed “equivalent latitude” (Butchart and Remsberg 1986; Allen and Nakamura 2003). Then FAWA is given by

*Q*–

*y*relation on the

*z*surface

*ζ*is relative vorticity,

*θ*is potential temperature,

*H*is a constant scale height. Now

*π*times) the absolute angular momentum of the zonal-mean flow (see NZ10 for details). Equation (4) captures the essence of conservative eddy–mean flow interaction: acceleration of the zonal-mean flow is achieved at the expense of wave activity. NZ10 also derives the evolution equation of

*z*surface without changing the enclosed areas. The corresponding flow

### b. New formalism: The local finite-amplitude wave activity

#### 1) Definitions

Although the FAWA formalism quantifies waviness in the PV contours and the associated mean-flow modification [see, for example, Solomon (2014) for stratospheric wave activity events], it is not suited to distinguish the longitudinal location of an isolated large-amplitude event such as blocking. To achieve this,

*Q*is displaced locally from

*η*can be multivalued in

*y*.) Now let

*η*. The eddy field is defined between

*y*; in other words, the eddy field is not defined globally as the total field minus the reference state, but it needs to be redefined for each

*y*. By definition,

*x*, we take the farthest crossings from equivalent latitude as

*z*surface, PV is generally greater on the northern side of the wavy contour than on the southern side such that

*y*and Eulerian (local) in

*x*. Notice that, since LWA vanishes at the nodes (i.e., crossing of the PV contour and equivalent latitude), it contains the phase structure of the waves in addition to the amplitude. In the small-amplitude, conservative limit, (12) becomes

#### 2) Local wave activity and PV gradient

*y*(see appendix A for the derivation):

*η*is multivalued, the sum of all values is used. Zonally averaging (17) and using (10) recovers (16). Differentiating this with respect to

*y*again yields

Polvani and Plumb (1992) discuss two regimes of wave breaking in the context of vortex dynamics: major Rossby wave breaking that disrupts the vortex dynamics and microbreaking that only sheds filaments and does not affect the vortex significantly [see also Dritschel (1988)]. In terms of LWA, a major breaking would satisfy (19) as well as a large amplification in LWA

#### 3) Local wave activity budget

*R*is gas constant, and

*x*component of (21) is

### c. Relationship to impulse-Casimir wave activity

*y*. It is defined as

*z*. ICWA obeys (Killworth and McIntyre 1985; Haynes 1988)

*y*–

*q*plane

Both wave activities obey similar equations [(20) and (24)], but while the ICWA equation is written entirely in terms of Eulerian quantities, the LWA equation involves line integrals and hence is Lagrangian in the meridional. A crucial difference arising from this is an extra meridional advection term

### d. Local nonacceleration relation

The nonacceleration relation (4) shows conservation of the sum of zonal-mean zonal wind and wave activity in a frictionless barotropic flow, but it does not tell whether the deceleration of the zonal-mean wind is because of growth of a localized wave packet or simultaneous growth of multiple wave packets over longitudes. To understand the dynamics of a localized phenomenon such as blocking, it is desirable to characterize eddy–mean flow interaction over a regional scale.

*x*. Thus, on short time scales

## 3. Numerical experiment

### a. Experimental setup

*ζ*is relative vorticity,

*J*is Jacobian,

*ψ*is streamfunction, and

*ν*is hyperviscosity, which we choose to damp the shortest resolved wave by a factor of 1/48 daily. We impose an initial zonal-mean flow as prescribed by HP87:

*m*,

*n*) = (4, 6) is added to break the zonal symmetry and allows merging of wave packets. Here,

*m*and

*n*are the zonal and total wavenumbers, respectively. The explicit form of

*ϕ*is latitude,

*λ*is longitude,

### b. Comparison between and

The overall flow evolution is similar to that in HP87: the wave packet initially located on the northern side of the jet axis splits into poleward and equatorward-migrating tracks, and, as they approach critical lines at the flanks of the jet, they produce wave breaking. The initial vorticity pattern consists of six pairs of positive and negative anomalies (Fig. 3, top), but their strengths are not symmetric because of the addition of small-amplitude, secondary wave

The snapshots of absolute vorticity, LWA

### c. Local negative correlation between and

For a zonal-mean state, the nonacceleration relation in (4) describes conservative eddy–mean flow interaction: *π*. We expect *χ* to be identical with the zonal mean. The question is whether the nonacceleration relation holds at a more regional scale *π* because of the presence of multiple waves, *x* = 0° and 60°E at 30°N and plotted as functions of time in the top panel of Fig. 5. This particular latitude is chosen because a prominent wave breaking occurs around here (Fig. 4).

The opposite tendency of the two quantities is evident, particularly during the early stage of simulation. Also plotted in the top panel are the sum

Figure 6 extends the above analysis to the entire latitude circle by showing the longitude–time (Hovmöller) cross sections (Hovmöller 1949) of *u* or *π*.)

We have repeated the analysis varying

## 4. Analysis of a blocking episode

Blocking is a phenomenon at midlatitudes in which a large-scale pressure anomaly remains stationary. The normal westerly winds in the mid- to upper troposphere are diverted meridionally along the blocking pattern, and the wind within the block is often replaced by easterlies. Lejenäs and Økland (1983) observed that blocking occurs at longitudes where the latitudinal average of the zonal wind at 500 hPa is easterly. Tibaldi and Molteni (1990) added an additional requirement that the average wind be westerly poleward of the block. Such description of blocking based on reversal of zonal wind is a kinematic statement. Given the potential of (36) to quantify the slowing down of the flow by finite-amplitude eddies, the formalism is well suited for identifying and investigating blocking events with meteorological data based on dynamics.

In this section, we explore the extent to which the dynamics of a real blocking episode may be characterized based on the conservation relation in (36). In particular, we will study the blocking episode that steered Superstorm Sandy to the U.S. East Coast during October 2012 with the LWA formalism. The interior and surface LWA as well as the zonal wind are evaluated from the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis product (ERA-Interim; Dee et al. 2011; http://apps.ecmwf.int/datasets/data/interim-full-daily/) at a horizontal resolution of 1.5° × 1.5°.

First, we evaluate PV from (3) on 49 equally spaced pressure pseudoheights, as described in Nakamura and Solomon (2010) (we assume that *H* = 7 km). Then, we compute

### a. Overview of zonal wind and LWA in northern autumn 2012

Longitude–time (Hovmöller) diagrams for the barotropic components of zonal wind ^{−1}, consistent with the phase speed of baroclinic waves (Williams and Colucci 2010). Thus, we believe that the streak pattern in LWA partly reflects the phase structure. However, the eastward migration of LWA is occasionally interrupted by large-amplitude, quasistationary features. A close correspondence is observed between these large LWA events and the reversal of zonal wind (i.e., negative

### b. Blocking episode around North American east coast during 27 October–2 November 2012

Now we focus on a single blocking episode that occurred during 27 October–2 November over the North Atlantic. (The longitudinal range of concern is marked by the black lines in Fig. 7.) This episode was characterized by a persistent blocking pattern in the mid- to upper troposphere and contributed to the steering of Superstorm Sandy at a right angle to the U.S. East Coast (Blake et al. 2013). Figures 8 and 9, respectively, show PV and the corresponding LWA

One might ask how the barotropic component (density-weighted vertical average) samples the vertical distribution of LWA associated with blocking. Figure 10 shows the vertical structure of

To examine the extent to which the local nonacceleration relation accounts for the simultaneous accumulation of LWA and deceleration of zonal flow, in Fig. 11 we show

There are several remarkable features from the plots. First, there is a strong negative correlation in the time series of ^{−1}. Since (36) states that this quantity is approximately invariant in time under conservative dynamics, it suggests that conservative dynamics cannot fully account for the occurrence of blocking. Given that the deceleration of the zonal flow has only half of the magnitude of the LWA anomaly, diabatic heating or other nonconservative processes are necessary to fuel the remainder of the LWA anomaly associated with this block. The discrepancy does not depend strongly on the averaging window, suggesting that the violation of the WKB condition is not the primary cause of the deviation from (36).

## 5. Summary and discussion

We have generalized the notion of FAWA introduced by NZ10 to LWA, a diagnostic for longitudinally localized wave events, and tested its utility in both a barotropic model and meteorological data. A significant advantage of LWA over the existing wave activity measures is that it carries over the nonacceleration relation of FAWA to regional scales, albeit within the WKB approximation. This explicitly attributes local deceleration of the zonal flow to accumulation of wave activity.

A robust negative correlation is found between

LWA dynamically connects the two criteria of blocking indices: 1) deceleration or even reversal of westerlies (Lejenäs and Økland 1983; Tibaldi and Molteni 1990) and 2) large amplitude of anomalies or gradient reversal in either geopotential height (at 500 hPa) (Barnes et al. 2012; Dunn-Sigouin and Son 2013) or potential temperature on constant potential vorticity surface (2 potential vorticity units) (Pelly and Hoskins 2003). Hoskins et al. (1985) suggests that meridional gradient reversal of potential temperature on a constant PV surface could imply a reversal of westerlies via the invertibility principle, but such a relation is not explicit. LWA can potentially serve as a blocking index because a large LWA will automatically lead to a significant deceleration of local zonal wind, to the extent that nonacceleration relation holds.

## Acknowledgments

Constructive critiques of the original manuscript by three anonymous reviewers are gratefully acknowledged. This research has been supported by NSF Grant AGS-1151790. The ERA-Interim dataset used in this study was obtained from the ECMWF data server (http://apps.ecmwf.int/datasets/data/interim-full-daily/).

## APPENDIX A

## APPENDIX B

### Derivation of (20)–(22)

Taking the time derivative and using the Leibniz rule and (9)

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