Local Finite-Amplitude Wave Activity as a Diagnostic of Anomalous Weather Events
Abstract
Finite-amplitude Rossby wave activity (FAWA) proposed by Nakamura and Zhu measures the waviness of quasigeostrophic potential vorticity (PV) contours and the associated modification of the zonal-mean zonal circulation, but it does not distinguish longitudinally localized weather anomalies, such as atmospheric blocking. In this article, FAWA is generalized to local wave activity (LWA) to diagnose eddy–mean flow interaction on the regional scale. LWA quantifies longitude-by-longitude contributions to FAWA following the meridional displacement of PV from the circle of equivalent latitude. The zonal average of LWA recovers FAWA. The budget of LWA is governed by the zonal advection of LWA and the radiation stress of Rossby waves. The utility of the diagnostic is tested with a barotropic vorticity equation on a sphere and meteorological reanalysis data. Compared with the previously derived Eulerian impulse-Casimir wave activity, LWA tends to be less filamentary and emphasizes large isolated vortices involving reversals of meridional gradient of potential vorticity. A pronounced Northern Hemisphere blocking episode in late October 2012 is well captured by a high-amplitude, near-stationary LWA. These analyses reveal that the nonacceleration relation holds approximately over regional scales: the growth of phase-averaged LWA and the deceleration of local zonal wind are highly correlated. However, marked departure from the exact nonacceleration relation is also observed during the analyzed blocking event, suggesting that the contributions from nonadiabatic processes to the blocking development are significant.
Waves play an important role of rearranging angular momentum in the atmosphere. This process is summarized by the generalized Eliassen–Palm (E–P) relation (Andrews and McIntyre 1976):
represents terms of third (and higher) order of 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.
In NZ10, FAWA 
is defined for latitude y, pressure pseudoheight z, and time t in terms of surface integrals of quasigeostrophic PV over two domains of equal area: 
and 
(see Fig. 1a). On a given z surface, 
is enclosed by a wavy PV contour of value Q, whereas 
is enclosed by a latitude circle at y. The requirement that the areas of 
and 
be identical enforces a one-to-one relation between Q and y for a given z. The latitude 
that encloses the same area as the PV contour of value Q is termed “equivalent latitude” (Butchart and Remsberg 1986; Allen and Nakamura 2003). Then FAWA is given by
is the length of the zonal circle. Note that under conservative quasigeostrophic dynamics, the Q–y relation on the z surface 
is independent of time because the wind that advects PV is divergence free and thus area preserving. In (2), PV is defined as (NZ10; Nakamura and Solomon 2010)
is the Coriolis parameter, ζ is relative vorticity, θ is potential temperature, 
is its global horizontal average, and H is a constant scale height. Now 
everywhere in 
, but 
includes some regions outside 
in which 
and excludes parts of 
(Fig. 1a), so the first integral in (2) is greater than the second, implying 
. The equal sign is achieved only when the PV contour coincides with the latitude circle. As noted by NZ10, in the small-amplitude, conservative limit 
converges to the more familiar expression for wave activity {i.e., 
}, where the overbar denotes zonal average and the prime denotes the departure from it.
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There are several advantages of 
over 
. First, while 
obeys a nonacceleration theorem that is accurate only up to the second order in wave amplitude 
, 
obeys an exact nonacceleration theorem under conservative dynamics. For example, for a frictionless barotropic flow (
),
in the presence of diffusion and forcing [their (24a)]. Second, the PV–equivalent latitude relation 
may be exploited to define a zonally symmetric, time-invariant “reference state.” It is a hypothetical distribution of PV that arises from “zonalizing” the wavy PV contours on the z surface without changing the enclosed areas. The corresponding flow 
and temperature field 
may be inverted from 
assuming the quasigeostrophic dynamics: for the barotropic case, simply 
. The notion of reference state may be generalized to a “slowly varying state” under nonconservative dynamics (Nakamura and Solomon 2010, 2011; Methven and Berrisford 2015). Third, unlike 
, which turns negative when 
, 
remains positive definite even when PV contours are overturned or fragmented, since 
is always positive by construction (Solomon and Nakamura 2012); 
is thus a more immediate measure of meridional displacement of a contour from zonal symmetry.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, 
needs to be generalized to a function of longitude as well. In the following, we assume that it is only the eddy properties that vary in longitude and continue to use 
, 
, and 
as a reference state to define the eddy fields. A zonally symmetric reference state may not reflect the zonally asymmetric nature of the time-mean flow, but it is a required construct for the conservation of wave activity. Keep in mind that, despite the enforced zonal symmetry, the reference state shares the same PV–area relation 
with the full wavy state, so it is strongly constrained to the actual climate state (Nakamura and Solomon 2010, 2011; Methven and Berrisford 2015).
Because of the waviness in the flow, the PV contour of value Q is displaced locally from 
to 
, where 
is defined positive northward. (As we will see below, η can be multivalued in y.) Now let 
or 
depending on the sign of η. The eddy field is defined between 
and 
as
is independent of 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,
follows most naturally by rewriting (2) as:
as
to denote wave activity that is a function of both longitude and latitude. It is evident from (11) that the zonal average of LWA recovers FAWA in (2). In practice, LWA is computed by evaluating (13) and (14). When there are multiple crossings of the PV contour with the meridian at a given x, we take the farthest crossings from equivalent latitude as 
and 
in (14) and use the PV constraint in (14) to sample the correct segments along the integral path. (Numerically, this amounts to a conditional box counting along the meridian.) Computation of 
is illustrated in Fig. 1b. On a given z surface, PV is generally greater on the northern side of the wavy contour than on the southern side such that 
in the red lobes and 
in the blue lobes. The line integral of 
over the red area and minus the line integral of 
over the blue are both positive, which makes 
a positive definite quantity. By construction, 
is Lagrangian (nonlocal) in 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
NZ10 shows in their (18) that FAWA bridges the Lagrangian- and Eulerian-mean PV via
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 
, whereas microbreaking would satisfy (19) without significant changes in 
.
The governing equation for LWA may be obtained by taking the time derivative of (12) (see appendix B for the derivation):
, R is gas constant, and 
is specific heat at constant pressure. The first term in the x component of (21) is 
at small amplitude and converges to 
. The second term, which is 
and represents the Stokes drift flux of 
, only becomes significant at finite amplitude. In this article, no further effort will be made to remove phase information from 
and fluxes other than averaging over a longitudinal window. [The methods described in Plumb (1985) and Takaya and Nakamura (2001) are not readily applicable to finite-amplitude wave activity.]Another well-known measure of local finite-amplitude wave activity is impulse-Casimir wave activity (ICWA), first introduced by Killworth and McIntyre (1985) and further developed by McIntyre and Shepherd (1987) and Haynes (1988). ICWA may be defined with respect to any zonally uniform, time-independent reference state in which PV (
) is a monotonic function of y. It is defined as
, and 
is an inverse function of 
for a given z. ICWA obeys (Killworth and McIntyre 1985; Haynes 1988)
is chosen to be identical with 
, then there is a close relationship between 
and 
. As illustrated in Fig. 2, on the y–q plane 
is given by the area bounded by 
, 
, and the curve 
(Fig. 2a), whereas 
is given by the area bounded by 
, 
, and the curve 
(Fig. 2b). When the eddy is of small amplitude [i.e., 
], these two areas are similar and both converge to (15). Where 
(nodes), they both vanish. However, once the PV gradient 
is reversed, 
becomes positive even at 
(Fig. 2c), whereas 
remains zero (Fig. 2d). In fact, 
tends to be greatest around the gradient reversal because both red and blue lobes in Fig. 1b (
) contribute to it. Consequently, 
emphasizes the region of wave breaking more than 
does, as we will see in the next section.
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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 
in (25), which does not have a counterpart in (21). The meridional advection of 
is absorbed in the movement of PV contour and does not appear in (21). The extra term in (25) prevents 
from possessing an exact nonacceleration theorem (NZ10).
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.
To formulate local eddy–mean flow interaction in a form analogous to (4), we start by taking the density-weighted vertical average of (20):
is the eddy geopotential. We also introduce local surface wave activity 
:
and its zonal average recovers the surface FAWA (NZ10; Wang and Nakamura 2015); 
obeys the equation
is barotropic streamfunction such that 
. Notice the zonal average of (32) gives
, denoted by 
:
would give
to be the wavelength, the right-hand side of (35) would be a small residual because of the slow modulation of wave properties in x. Thus, on short time scales
because of the additional meridional flux term in (24). In the next section, we will compare LWA and ICWA using idealized numerical simulations in which finite-amplitude Rossby waves are allowed to interact with shear flow on a rotating sphere. The extent to which the above conservation law for LWA is satisfied will be examined. In section 4, the LWA diagnostic is applied to meteorological reanalysis data to identify and analyze an atmospheric blocking event.The utility of the LWA diagnostic will be tested in a barotropic decay simulation of finite-amplitude Rossby waves as described by Held and Phillipps (1987, hereafter HP87). The governing barotropic vorticity equation reads as follows:
is the Coriolis parameter (
is the rotation rate of the sphere), 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:
of the form in HP87, which is a Gaussian wave packet meridionally centered at 
N with zonal wavenumber 
. Then another wave with (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 
(see Fig. 3, top) is
, 
, 
N, and 
. We discretize the equation with a standard spectral transform method truncated at T170 on a Gaussian grid of resolution 512 × 256. The Adams–Bashforth third-order scheme (see Durran 2013, chapter 2.4) is used to integrate the equation with a time increment of 
until the major wave packet decays completely. The computation of 
and 
is implemented on instantaneous snapshots of the vorticity field obtained from the simulation. Since the model is barotropic, the third dimension in the fluxes in (21), (22), (25), and (26) is ignored, and potential temperature and surface LWA 
are set to zero. The local nonacceleration relation in (36) is simplified to
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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 
. As the wave packet begins to separate meridionally, six positive vorticity anomalies move northward, whereas six negative anomalies move southward, and by day 3, the vorticity contours begin to overturn at the flanks of the jet. (Here, anomalies are defined as departures from the zonal mean of the initial state; see Fig. 3, bottom.)
The snapshots of absolute vorticity, LWA 
, and ICWA 
, are shown for days 3 and 6 in Fig. 4 over the Northern Hemisphere. The positive anomalies form isolated vortices around 50°N, whereas the negative anomalies develop marked anticyclonic tilt at the equatorward flank of the jet (Fig. 4, top left). Both 
and 
identify large vorticity anomalies, but there are substantial differences between their spatial distribution. The LWA emphasizes the four largest positive anomalies, although they are shifted and elongated poleward from the actual locations of the vorticity anomalies (Fig. 4, middle left). This is a nonlocal effect of 
: the isolated vortices are indeed associated with a higher equivalent latitude. The ICWA also picks up the isolated vortices, but they tend to be much more compact and intense than 
. Also, the structure of 
around the negative anomalies appears more filamentary than 
. Part of this difference is because, as explained in the previous section (in Fig. 2d), 
tends to suppress wave amplitude in the region of reversed vorticity gradient; for example, the value of 
drops from a maximum to zero to the north and south of isolated vortices. By day 6 (Fig. 4, right), a pair of vortices start to merge poleward of the jet around 10°–110°E and 70°–170°W. In the 
plot, the merging vortices appear as one bulk structure, whereas in 
they are more fragmented. On the equatorward flank of the jet, wave breaking causes the negative vorticity anomalies to roll up. The plot of 
captures these emerging vortices faithfully; but 
is highly filamentary around them. Similar filamentary structures of 
have been observed in previous analyses related to baroclinic life cycles and Rossby wave breaking (Magnusdottir and Haynes 1996; Thuburn and Lagneau 1999).
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and 
For a zonal-mean state, the nonacceleration relation in (4) describes conservative eddy–mean flow interaction: 
accelerates at the expense of 
, and vice versa; thus their variation is antiphase. The value of 
is constant in time if the dynamics is conservative, so any changes in 
are due to nonconservative processes; in the present case, they represent damping of FAWA through vorticity mixing (enstrophy dissipation by hyperviscosity). Since the initial condition in (39) creates interference of zonal wavenumbers 4 and 6, the resultant flow has a zonal periodicity π. We expect 
[cf. (34)] for any physical quantity χ to be identical with the zonal mean. The question is whether the nonacceleration relation holds at a more regional scale 
as in (36). Although there is no strict periodicity below π because of the presence of multiple waves, 
still remains a dominant zonal wavenumber, so 
would be a reasonable choice of the averaging window. The values of 
and 
are computed between 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).
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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 
and 
. The zonal averages of the two quantities are identical. The slow variation of 
reflects rearrangement of angular momentum by vorticity mixing, which is not included in (4). The sum 
follows 
generally well, suggesting that the long-term changes in 
are due to mixing. The early disagreements are largely due to periodic modulation of 
by waves with wavelengths greater than 
, but the range of fluctuation in 
is generally smaller than that of 
or 
alone, attesting to the overall validity of (36). Similar analysis is performed for 
in the bottom panel of Fig. 5. Compared to 
, 
varies much less, and its anticorrelation with 
is far less evident. Accordingly, the sum of 
and 
varies more in time. This demonstrates that the local nonacceleration relation in (36) is generally not applicable to 
.
Figure 6 extends the above analysis to the entire latitude circle by showing the longitude–time (Hovmöller) cross sections (Hovmöller 1949) of 
, 
, and 
anomalies (departure from the time mean) at 30°N (
). Because of the averaging, the fields are devoid of zonal wavenumber 6, the predominant structure in the unfiltered data. Instead, the analysis picks out the emerging wavenumber 2, which modulates the averaged quantities. The negative correlation between 
and 
is again evident, and it holds not only in time but also in longitude (particularly strong in the early stage). This is important because it suggests that the nonacceleration relation in (40) is applicable regionally. On the other hand, (40) is not perfect: 
shows a significant residual in the bottom panel. As mentioned above, this is partly because of nonconservative effects (vorticity mixing). It also contains a wavenumber-2 component, which represents group propagation of 
expressed by the right-hand side of (35). Although the amplitude of this variation is smaller than the amplitude of u or 
, its nonnegligible magnitude suggests that the scale separation required for (40) is insufficient. (In the present case, the wavelength of the dominant wave is 
, whereas the packet size is π.)
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We have repeated the analysis varying 
(
and 
) and found (not surprisingly) that 
deviates from 
more when we reduce 
further. Arguably, this simulation is a special case in which the wave spectra are highly discrete. In a sense, it is even less obvious how best to choose an optimal 
when the waves have broader spectra. We will see in the next section that, dealing with the real atmospheric data, horizontal averaging may actually be forgone.
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 
from (13) with PV south of 7.5°N excluded; 
is computed from (30), but we have replaced the surface potential temperature with the potential temperature at 866 hPa to avoid the nonquasigeostrophic effects in the boundary layer.
Longitude–time (Hovmöller) diagrams for the barotropic components of zonal wind 
, LWA 
, and their sum at 42°N during this season are shown in Fig. 7. Notice that in this analysis we are not using any horizontal average defined by (34). Indeed, the prevalent short streaks in LWA (middle panel) suggest an average eastward migration of LWA at about 11 m s−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 
) in the left panel, although the magnitude of fluctuation in LWA is about twice as large as that of the zonal wind (notice the different color scales for the two quantities). The fluctuation of their sum (Fig. 7, right) has a smaller variation than 
. The simultaneous growth of LWA and the deceleration of zonal flow are characteristic of blocking. Remarkably, neither does the unfiltered phase signal hinder the detection of blocking, nor does removal of the phase by averaging improve the result significantly. It appears that LWA has no problem detecting the packet structure of blocks without regional averaging (34). Part of the reason is that the last two terms in (32) nearly cancel in geostrophic balance and that the vertical averaging in the other right-hand side terms, when the phase surfaces are tilted vertically, achieves the same effect as the phase averaging.
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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 
at 240 hPa. There is an intrusion of low-PV air poleward at 40°N, 290°E, which remains stagnant longitudinally (relative to other eastward-migrating features) for 2 days and eventually splits into two asymmetric vortices. The smaller vortex that moves westward accompanied Sandy onshore. The location and magnitude of the block are well captured by high values of 
in Fig. 9.
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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 
(left) and density-weighted LWA (
; right). Even though the pattern of blocking is apparent in 
only at the upper levels (i.e., 300–150 hPa), density weighting indeed brings out a vertically coherent structure of high LWA, as shown in Fig. 10 (right). Thus, what we observe in Fig. 7 represents a persistent block affecting an entire troposphere and not just upper levels, both in terms of the accumulation of LWA and the deceleration of the flow.
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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 
(red), 
(blue), and their sum (green) averaged longitudinally over 270°–330°E (the longitudes bounded by the black lines in Fig. 7) at different latitudes within the meridional extent of the blocking episode. Here, 
denotes departure from the seasonal average. This graph is analogous to Fig. 5 for the barotropic decay simulation. The correlation coefficient of the time series of 
and 
throughout the analysis period is displayed at the top-left-hand corner of each plot.
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There are several remarkable features from the plots. First, there is a strong negative correlation in the time series of 
(red) and 
(blue), clearly indicating the antiphase covariation of the two quantities expected from the nonacceleration relation. This relation is particularly visible during the block (27 October–2 November) when the amplitude of the wave is large. Second, the 60° longitudinal average of 
(green) weakly oscillates about zero, except during the time of blocking formation, when LWA grows large. Its peak value exceeds 20 m s−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).
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 
and 
in both a simulated wave breaking and an observed blocking event, suggesting that the quasiadiabatic eddy–mean flow interaction is indeed of leading-order importance in these weather events. Nevertheless, the variation of 
during the blocking event is about twice as much as that of 
, which implies that not all LWA growth is accounted for by the simultaneous deceleration of the zonal flow. Diabatic and other nonconservative processes are responsible for half the budget of the 
anomaly. This perspective is consistent with a recent study based on the formalism by NZ10 (Wang and Nakamura 2015) that shows that the variability of the eddy-driven jet in austral summer is largely dictated by conservative dynamics of wave–mean flow interaction but moderated by strong thermal damping of surface wave activity. Strong damping of (negative) 
would render the values and variability of 
higher than those expected under the adiabatic condition, consistent with our analysis. The precise role of nonadiabatic effects on blocking formation will be a subject of subsequent work.
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/).
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From (12),
Taking the time derivative and using the Leibniz rule and (9)
is
Note that the spatial derivative of the eddy quantities is taken with respect to the coordinates 
, and we used nondivergence of 
. Substituting (B3) into (B2) yields [with repeated use of the Leibniz rule and (9)]
