Editorial Type:
Article
Article Type:
Research Article

Local Wave Activity and the Onset of Blocking along a Potential Vorticity Front

Noboru Nakamura Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois

Search for other papers by Noboru Nakamura in
Current site
Google Scholar
PubMed Close
and
Clare S. Y. Huang Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois

Search for other papers by Clare S. Y. Huang in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Jul 2017
DOI:
https://doi.org/10.1175/JAS-D-17-0029.1
Page(s):
2341–2362
Full access

Abstract

Interaction between a train of transient waves and a diffluent westerly jet is examined using a regional quasigeostrophic equivalent barotropic model with a (nearly) binary potential vorticity (PV) distribution. Unlike most previous studies, but consistent with the observed extratropical tropopause, cross-stream variation in the layer thickness is allowed to contribute to the discontinuity in PV. In all cases examined, short (i.e., barotropic) edge waves are continuously forced in the upstream, then migrate downstream, and eventually exit the domain. A quasilinear 1D theory based on the conservation of local wave activity predicts that no steady wave train can be maintained where the westerly zonal flow is decelerated below one-half of the initial value, at which point the wave envelope develops a migratory shock analogous to the Lighthill–Whitham–Richards traffic flow problem. Fully nonlinear high-resolution 2D calculations show that the wave train indeed undergoes a significant transformation once the zonal flow along the jet axis is decelerated below the threshold. The subsequent flow evolution depends on the nature of the discontinuity in the basic-state PV. When the discontinuity is entirely due to the vorticity profile, waves are compressed and partially deflected sideways but no complete blocking occurs. When the discontinuity in PV is augmented by the layer thickness variation, the incident wave train is blocked and split into two tracks at the stagnation point, eventually leading to a formation of a modon-like vortex pair, reminiscent of an atmospheric blocking. Implications for low-frequency variability of the atmosphere are discussed.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Noboru Nakamura, nnn@uchicago.edu

Abstract

Interaction between a train of transient waves and a diffluent westerly jet is examined using a regional quasigeostrophic equivalent barotropic model with a (nearly) binary potential vorticity (PV) distribution. Unlike most previous studies, but consistent with the observed extratropical tropopause, cross-stream variation in the layer thickness is allowed to contribute to the discontinuity in PV. In all cases examined, short (i.e., barotropic) edge waves are continuously forced in the upstream, then migrate downstream, and eventually exit the domain. A quasilinear 1D theory based on the conservation of local wave activity predicts that no steady wave train can be maintained where the westerly zonal flow is decelerated below one-half of the initial value, at which point the wave envelope develops a migratory shock analogous to the Lighthill–Whitham–Richards traffic flow problem. Fully nonlinear high-resolution 2D calculations show that the wave train indeed undergoes a significant transformation once the zonal flow along the jet axis is decelerated below the threshold. The subsequent flow evolution depends on the nature of the discontinuity in the basic-state PV. When the discontinuity is entirely due to the vorticity profile, waves are compressed and partially deflected sideways but no complete blocking occurs. When the discontinuity in PV is augmented by the layer thickness variation, the incident wave train is blocked and split into two tracks at the stagnation point, eventually leading to a formation of a modon-like vortex pair, reminiscent of an atmospheric blocking. Implications for low-frequency variability of the atmosphere are discussed.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Noboru Nakamura, nnn@uchicago.edu
Keywords: Blocking; Conservation equations; Rossby waves; Wave breaking; Quasigeostrophic models; Intraseasonal variability

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

We first examine a small-amplitude, equivalent barotropic dynamics of edge waves propagating on a steady PV front, notionally representing the extratropical tropopause region (Harvey et al. 2016). Consider a semi-infinite f plane . Quasigeostrophic PV in the steady basic state is assumed to be uniform in but discontinuous at :
e1
where is a constant Coriolis parameter; is assumed; is the Rossby radius, with gravitational acceleration g and average layer thickness H; is horizontal Laplacian; and is the basic-state streamfunction. Distinct from most previous studies, we allow for a meridional variation of layer thickness through bathymetry to contribute to the PV discontinuity (a crude representation of the meridional distribution of isentropic density across the tropopause). The rationale and significance of this term will be discussed in section 3. Furthermore, we assume that the basic-state zonal velocity at y = 0, , is everywhere positive but a slowly varying function of x. A specific form of to achieve this while enforcing (1) will be also given in section 3.
Now a small-amplitude wavy perturbation is introduced by displacing the PV front meridionally . This induces fluid motion governed by
e2
and
e3
where is the perturbation streamfunction and is the jump in the meridional gradient of across the PV front. Note that (2) is obtained by integrating the linearized PV equation across the PV front over an infinitesimal distance (Swanson et al. 1997; Swanson 2000). The displacement and streamfunction are related through (Swanson et al. 1997)
e4
Since U varies only slowly in x, (2) admits a steady WKB solution in the form
e5
where (ε ≪ 1) is the zonal coordinate of slow variation and
e6
is the effective horizontal wavenumber. Figure 1 shows a schematic of the wave structure. Note that for a steady wave the frequency ω is independent of X since ω and k are related through a phase function such that
e7
so . Also 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)
e8
The phase and group velocities are
e9
and
e10
respectively. Finally, substitution of (4) and (5) in (2) yields
e11
which is also valid if the flow is slowly evolving in time.
Fig. 1.
Fig. 1.

Structure of a slowly modulated edge wave on a PV front. PV is in the pink region and in the blue region. Although PV is discontinuous at the interface, it is given a notional value of there. The PV front is initially zonally straight, located at . Local wave activity is (minus) the line integral of perturbation PV between and the meridional displacement . The phase average of LWA is , where is the combined area of the two stippled lobes divided by the local wavelength . The exchange of PV across leads to deceleration of the local zonal flow along . See text and the appendix for details.

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

In what follows we will focus on high-frequency waves that satisfy kLD ≫ 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:
e12
Notice that in this limit the group velocity equals the local basic-state zonal wind and does not depend on the properties of the wave.
It is well known (Swanson et al. 1997) that the above WKB solution satisfies the conservation of wave action [Bretherton and Garrett 1968, Eq. (1.9)]
e13
e14
where we used (12) in (13); the angle brackets in (14) denote a local phase average, and the last identity uses (4) and (11). If is a locally sinusoidal function with wavenumber and amplitude , then (14) becomes
e15
Because ω is independent of X for a steady wave, (12) gives . It thus follows from (13) that and then (15) implies
e16
Therefore, both the amplitude and wavenumber of the displacement increase as the basic-state U decreases. Meanwhile, from (11) and (12),
e17
Thus, the amplitude of is proportional to , which, according to (16), is constant with 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

So far we have considered linear dynamics in which waves and basic-state flow do not interact. However, if waves with finite amplitude rearrange momentum in the flow and when a steady state is reestablished, the zonal flow at in the new state will be different from the initial condition . Here is the local phase average of zonal velocity u that excludes fast oscillation associated with waves. In the presence of finite-amplitude waves it is that should be used for the dispersion relation [(12)–(16)] instead of U. To understand the change from U to , it is useful to invoke the conservation of LWA (HN16), which provides additional insight on the wave–zonal flow interaction. HN16 defines LWA, denoted in this article by , as the amount of meridional displacement of PV from zonal symmetry at each longitude. For the PV front under consideration (see the appendix),
e18
The phase average of is
e19
where the last equality assumes a local sinusoidal function with amplitude for . Note that equals the combined area of the two stippled lobes in Fig. 1, enclosed between the wavy PV front and the line , divided by the local wavelength .
In the appendix it is shown that in the quasilinear WKB limit is governed by
e20
The first term in the second expression is (minus) the phase-averaged meridional vorticity flux across , whereas the second term is the convergence of the zonal advective flux of LWA along . Notice that for the PV front problem is a first-order quantity in wave amplitude [(19)] unlike wave action , which is second order [(14)], and the effective transport velocity of in (20) is the phase velocity instead of the group velocity.
The corresponding equation for the phase-averaged zonal flow at is (see the appendix)
e21
Here is the part of the local zonal flow induced by the waves; that is, . Notice the similar forms of (20) and (21), both carrying the same vorticity flux term with opposite signs and a zonal flux convergence term.
Suppose initially there are no waves and —that is, everywhere—and at some point in a remote upstream region a wave forcing is switched on through the vorticity flux term. Since the vorticity flux terms dominate in the forcing region, and since they have opposite sign in (20) and (21), it suggests as waves grow in that region. Now add (20) and (21) together to obtain
e22
The solution to (22) is
e23
because it satisfies the initial condition everywhere and also the boundary condition just outside the source region, as suggested above, for all time. Since , this means
e24
This is a quasilinear local nonacceleration relation for the PV-front problem, a variant of the results of Charney and Drazin (1961) and Nakamura and Zhu (2010) (see HN16 for more details). Note that even though (24) applies locally, its derivation relies on nonlocal analysis because of the zonal flux terms in (20)–(22). In fact, the zonal flux convergence dominates the vorticity flux in (20) as long as c is close to . This means that the last expression of (20) does not apply to the source region in which the vorticity flux is assumed to be dominant.
In a steady state in which waves are present, (20) may be rewritten as
e25
where we used (12) replacing U with . Since ω is constant in a steady state, this leads to
e26
where (24) was used. Therefore, in the steady state the downstream advective flux of LWA is independent of X:
e27
Since in the steady state the total LWA flux matches the production rate of LWA in the source region, may be determined from that rate together with ω and . Given and the initial basic-state flow , (27) forms a quadratic equation for with two possible roots:
e28
if
e29
The two roots are possible steady-state values of at X. It turns out that the plus root in (28) is unstable and thus unrealizable: if increases further, the flux [the left-hand side of (27)] decreases below . Assuming the flux is in the upstream, this makes the flux decrease downstream, causing convergence and further increasing . On the other hand, for the minus root, an increase in would make the flux greater than and induces divergence and hence decreases , causing the solution to be stable. (The same conclusion holds when decreases below the threshold.) When (29) is violated, there is no real root for . In this case LWA cannot maintain a steady state. For a given initial zonal wind there is a maximum LWA flux (=U2/4) 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, . In other words, once the zonal flow is decelerated by one-half, LWA becomes unsteady. Modification of the flow by the waves [(24)] is essential to this threshold behavior. Without the wave–flow interaction (i.e., the quadratic term), there is always a stable steady solution as long as and are nonzero and of the same sign. It is also important to note that the threshold behavior occurs before the flow is decelerated to zero (i.e., before the critical point is reached).

b. Transient behavior

To investigate the transient behavior of the wave train when the above threshold is reached, we solve numerically the full conservation equation of LWA (20)
e30
together with the wave equation [(2)] in the barotropic limit . Note that in (30) has been rescaled to . Although in the time-dependent problem 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 . The initial zonal flow U is specified as
e31
and time, length, and velocity scales of zonal advection [, , and , respectively] are used to nondimensionalize (30) and (31):
e32
and
e33
where . The asterisk denotes nondimensional quantities. The corresponding nondimensional form of (2) is, after substituting (4),
e34
where has been replaced by to incorporate the wave–flow interaction. In the calculations below, we choose , , and . This choice allows substantial streamwise variation of the zonal flow while assuring that the wavelength is much shorter than the scale of the flow variation, consistent with the WKB theory. The initial zonal flow is maximal (=1.46) at and 2π and minimal (=0.54) at . Waves are forced at as
e35
in (32) and
e36
in (34) to allow for a gradual switch on. For both and a radiation condition (Orlanski 1976) is assumed at the downstream boundary (x* = 2π) 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 , where is the grid size and is the time step interval.] The necessary inversion from to in (2) is trivial with (4), (17), and (12).

Figure 2 depicts the numerical solutions for at four different stages of evolution (t* = 3.3, 6.6, 9.9, 13.2). With this choice of we have and (29) is satisfied everywhere. For each instance the top panel shows the phase structure of (black curve) and its amplitude envelope (red) computed as . The predominant wavenumber in the upstream is determined by (12) as or a wavelength of 0.54. The middle panel shows the initial flow (gray), (red), (black), and the theoretical prediction of the asymptotic zonal flow (green). The steady solution of obtained from the minus root of (28) is used to compute the green curve. Notice that the predicted (green) is everywhere above (red) in the middle panel, consistent with (29). The bottom panel shows the downstream advective LWA flux . Eventually the wave envelope, the zonal flow, and the LWA flux all approach a steady state (t* = 13.2). The amplitude and wavenumber of the wave train are modulated according to (16): they are both greatest in the region where is smallest, yet the LWA flux is constant throughout the channel. The actual zonal flow (thin back) at the final stage (t* = 13.2) is in excellent agreement with the theoretically predicted asymptotic zonal flow (green).

Fig. 2.
Fig. 2.

Longitudinal structure of a 1D transient wave train traversing a zonally varying flow at different instances. Moderate wave forcing . (top left) , (top right) , (bottom left) , and (bottom right) . For each instant, (top) the meridional displacement of the PV front (black) and its envelope (red); (middle) the initial zonal flow (gray), one-half of the initial zonal flow (red), asymptotic zonal flow predicted by (28) and (24) (green), and the actual zonal flow (thin black); and (bottom) the zonal advective wave activity flux . To relate to the midlatitude atmosphere, the following dimensional scaling may be useful: channel length , , , , and the display time interval ~2 days. See text for details.

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

Figure 3 shows an analogous result for with other parameters unchanged from Fig. 2. In this case and (29) is first violated at . This is indicated by a break in the green curve in the middle panels, at which point the predicted equals . Initially the behaviors of the wave train, zonal flow, and the LWA flux are qualitatively similar to Fig. 2 (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 The stepwise loss of the LWA flux across the discontinuity is balanced by the overall growth of LWA, whereas the nearby spike in the flux is compensated by the loss of LWA due to numerical diffusion. The wave envelope predicted by (32) traces the crests remarkably well even after the discontinuity forms. Although the zonal flow is significantly decelerated by the enhanced wave activity, it manages to remain positive over the entire domain: no reversal of the zonal flow occurs.

Fig. 3.
Fig. 3.

As in Fig. 2, but with a stronger wave forcing . The asymptotic zonal flow predicted by theory (green) is only displayed within the range that (29) is fulfilled. The wave envelope starts to grow at at the break of the green curve and migrates upstream with time.

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

It is noteworthy that (30) is a close relative of the Lighthill–Whitham–Richards equation for the traffic flow problem (Lighthill and Whitham 1955; Richards 1956) whose solutions are well studied [LeVeque (2002), section 11.1; Treiber and Kesting (2013), chapter 8]. In that context would be the traffic density and would be the traffic speed. The traffic speed depends not only on the imposed speed limit but also on the traffic density—heavy traffic naturally makes the flow slower. The solution to (30) develops a shock (a sudden “traffic jam”) where if there. In the present case so this occurs at when is a decreasing function of X, but a shock can form in a broad class of as long as it satisfies the aforementioned condition. Once the shock forms, it migrates with the Rankine–Hugoniot speed:
e37
where the superscript plus and minus signs denote values on the downstream and upstream sides of the shock, respectively. Figure 4 demonstrates that this weak solution of the Riemann problem, when carefully evaluated, reproduces the observed migration of the shock in the numerical solution in Fig. 3 accurately.
Fig. 4.
Fig. 4.

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 that maximizes the difference between and . Here is the grid size and 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 between 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.

The initial basic state is a (near) steady-state solution of the PV equation and consists of a zonally symmetric component plus a 2D component. The zonally symmetric part of streamfunction is given in a dimensional form by
e38
Here represents a slightly smoothed cusp jet with the axis at . The constants A and B are determined such that and its first derivative are continuous at [see (47) below]. The concentrated flow curvature around creates a localized PV gradient.
Usually in a problem like this, the concentrated flow curvature (vorticity gradient) is the sole contributor to the (near) discontinuity in PV. In reality, however, vorticity gradient is not the leading contributor to the PV gradient in the extratropical tropopause region. Figures 5a–c show the seasonal climatology of relative circulation, isentropic density (layer thickness), and Ertel PV (Hoskins et al. 1985) as a function of equivalent latitude on the 340-K isentropic surface for December–February. ERA-Interim (Dee et al. 2011) for 1979–2014 was used to compute them. (See the figure caption for the method of computation.) The 340-K isentropic surface intersects the tropopause in the midlatitudes, where both relative circulation and the gradient of PV are maximized (Figs. 5a and 5c). The use of equivalent latitude preserves the sharpness of the jets, as evident in the comparison with the zonal-mean zonal circulation in Fig. 5a. Furthermore, isentropic density (layer thickness) varies greatly from pole to equator (Fig. 5b), as it reflects the transition from the (inverse of) stratospheric to tropospheric static stability. [See also Fig. 7 of Wang and Nakamura (2016) for the observed potential temperature distribution around the extratropical tropopause in the Southern Hemisphere.] Figure 5d describes the PV gradient (solid curve) and the contribution from the density variation to it (thin solid curve). One can see that more than half of the PV gradient across the tropopause is due to the density variation and the contribution from vorticity gradient is smaller. Table 1 summarizes the magnitude of the cross-tropopause PV variation on the 340-K surface and the relative contribution of the thickness variation to it for both hemispheres and different seasons. The ratio in the last column remains consistently above 50% and reaches as high as 65% in summer. To the extent that the PV discontinuity in the model represents the PV gradients around the extratropical tropopause, it should incorporate the effects of thickness variation. To achieve this within the quasigeostrophic framework, we allow the rest layer thickness to vary modestly across the jet through bathymetry in (1):
e39
[Note that the isentropic gradient of the Ertel PV multiplied by the mean isentropic density is analogous to the gradient of the quasigeostrophic PV; see, for example, Fig. 2 of Nakamura and Solomon (2011).] The bathymetry is assumed static since the edge waves on the PV front are much shorter than the Rossby radius and do not affect the layer thickness. Together with (38), (39) satisfies a modified form of (1):
e40
assuming . In the limit of vanishing δ, the PV jump has contributions from both the jet and from the change in the layer thickness:
e41
The relative contribution of the thickness and the vorticity variations to the PV jump is determined by the ratio of the fractional change in the thickness to the Froude number of the jet —see (48) below. A slight difficulty with this formulation is that the Taylor–Proudman theorem discourages barotropic fluid to cross steep bathymetry and thus hinders the lateral wave motion. We simply assume that wave forcing provides the necessary push to move the fluid across the barrier and just solve the PV equation using (40); that is, (39) is only implicit through the definition of the background PV. It turns out that the additional contribution to the PV discontinuity from the thickness variation is important for the generation of blocking in this model.
Fig. 5.
Fig. 5.

December–February climatology of the Lagrangian-mean state on the 340-K isentropic surface as functions of equivalent latitude . Here equivalent latitude is defined as the latitude that encloses the same isentropic layer mass on the poleward side as the given PV contour, following Nakamura and Solomon (2011). (a) Relative circulation divided by , where a = 6378 km is the planetary radius. Note , where 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 . For a comparison, the Eulerian zonal-mean zonal wind is shown in the dashed curve. (b) Isentropic density , where 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 , where is the Lagrangian-mean absolute vorticity. (d) Thick solid curve: meridional gradient of the Ertel PV . Thin solid curve: contribution to the meridional PV gradient from the thickness variation . The difference between the two is the contribution from the vorticity gradient . All these quantities are evaluated instantaneously and then averaged over the three months. Based on 1979–2014 ERA-Interim product Dee et al. (2011). See text for details.

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

Table 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).

Table 1.
In addition to , the basic-state streamfunction includes a 2D component :
e42
makes no contribution to PV but gives rise to a streamwise variation in the zonal flow along the PV front identical to the 1D problem (31):
e43
Note that diverges at large y, but we will consider the dynamics only in a finite domain in which is well bounded. The term may be thought of as a stationary wave being forced in remote locations outside the domain, and is a steady solution of the nonlinear PV equation
e44
in the limit of vanishing δ, where J is the 2D Jacobian operator. For a transient problem we solve the full PV equation
e45
from the initial condition and on a rectangular domain, with wave forcing slowly switched on in the upstream. Here and are the full streamfunction and PV satisfying
e46
and ν is hyperviscosity to diffuse PV at small scales.
Again we nondimensionalize time, horizontal length, and velocity using , , and , respectively. Furthermore, we assume —namely, the zonal scale of the background stationary wave is comparable to the Rossby radius—whereas that of the incident transient waves is sufficiently smaller. Therefore the Rossby radius affects the geometry of the initial flow but not of the transient waves. After the nondimensionalization, (38)–(41) become, respectively,
eq1
e47
e48
and
e49
e50
Thus represents the contribution from the thickness variation to the PV jump . Meanwhile, the nondimensional form of (42) is
e51
whereas (43) becomes identical to (33).
The nondimensional form of (45),
e52
is discretized on a rectangular domain with unstaggered grids, using a flux form of second-order finite difference. The physical domain size in and is set to 2π and 0.6π, 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 with a damping rate linearly increasing from zero in the interior to at the boundaries of the computational domain. It is designed to absorb and damp wave energy injected from upstream (and possibly scattered back internally) and to eliminate spurious reflections by the boundaries. We prefer the sponge layer to a uniform domainwide damping so as to keep the interior dynamics close to conservative, since the theory is based on the conservation of LWA. The discrete form of (52) is integrated in time from the basic state using the third-order Adams–Bashforth method (Durran 1991) for time differencing. Transient waves are forced by a periodic meridional displacement of the PV front at (just outside the sponge layer):
e53
where is identical with (36). The radiation boundary condition is applied to (although this is not too critical given that the waves are strongly damped by the sponge layer), whereas is fixed at the initial values on the boundaries of the computational domain. At each time step the interior is inverted from with the nondimensional form of (46),
e54
using a direct method. In what follows we fix the following parameters throughout:
e55
Unless otherwise stated, the domain is discretized using 2401 × 721 grids in and including the sponge layer. The flow profile along the axis of the jet and the forcing frequency of the waves are identical to the 1D problem discussed earlier. The key parameters to be varied are the wave forcing amplitude and the cross-stream thickness variation (and thus the PV jump ).

a. Weak forcing without thickness variation

First, we assume that there is no thickness variation —namely, PV varies solely as a result of flow curvature . Figure 6 samples snapshots of PV (left) and streamfunction (right) at different stages with a weak wave forcing . The choice of parameters closely mirrors that of the first 1D experiment in the previous section (Fig. 2). The transient waves are visible in the displacement of PV front. As the zonal wind at the wavefront decreases, the wavenumber and amplitude of the waves increase as expected. Because of the meridional shear in the flow and because of their small scales, the PV anomalies are deformed passively. The lobes of PV are progressively tilted backward downstream, although straining by the shear prevents them from becoming unstable. The envelope of the wavy PV contours remains well contained inside the westerly region demarcated by the separatrices of streamfunction, noted by the red curves in each panel of Fig. 6. (The separatrices are defined by the set of contours of that intersect at the stagnation points.) As streamfunction barely changes its structure, the separatrices also remain nearly stationary.

Fig. 6.
Fig. 6.

PV and streamfunction at different stages of the 2D numerical experiment with for (top)–(bottom) t* = 3.3, 6.6, 9.9, 13.2. The domain shown excludes the sponge layer. (left) . In the green and blue areas is 1 and −1, respectively, and the PV front is traced by three contours: . [A constant background value is subtracted from .] (right) . The contours for are drawn between −0.37 and +0.37 with an interval of 0.02. Red and blue colors indicate positive and negative values of . The thin red curves in each panel show the separatrices of . See text for details.

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 . Consistent with the prediction, the zonal flow (black curve) is decelerated but remains higher than one-half of the original value (red curve) at all . The dashed curve in the left column is the sum of and . If (24) is obeyed, it should coincide with the gray curve []. The small deficit reflects the loss of due to dissipation of PV filaments by hyperviscosity (mixing). The LWA flux increases downstream up to , where the divergence balances the forcing (applied at ). Further downstream, the flux decays slowly with unlike the 1D problem in which it remains flat (Fig. 2). This slow decay and the associated weak flux convergence are balanced by the loss of LWA due to mixing. The decay rate increases downstream as more PV filaments are shed and LWA is lost to mixing.

Fig. 7.
Fig. 7.

(left) The zonal-flow profile and (right) the zonal advective LWA flux at corresponding to Fig. 6 . (left) The solid black curve shows instantaneous , the thick gray curve is the initial , the thin red curve is one-half of the initial , and the dashed curve is . The value of is computed by taking the finite difference of in the meridional direction. A running mean over 200Δx* 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 . This corresponds to the second 1D experiment in section 2, in which (29) was violated around . The results are shown in Figs. 8 and 9. The wavefront slows down as it reaches the diffluent region; the PV contours are compressed into bow-shaped filaments, gradually increasing the wavenumber and the amplitude of meridional displacement (Fig. 8; ). The upstream LWA flux in Fig. 9 proves somewhat smaller than the 1D case (1.14 × 10−1) owing to the smoothing of the PV front, and accordingly (29) is first violated farther downstream (, ; Fig. 9). Once the zonal flow is decelerated below one-half of the initial value, the wave envelope undergoes a substantial deformation: it flares sideways and partially exits the domain through the side boundaries around (Fig. 8; ). (Changing the meridional width of the domain did not affect the solution.) In the region where (29) is violated (2.8 < x* < 3.2) the zonal LWA flux decreases rapidly downstream, qualitatively similar to Fig. 3 (Fig. 9; ). In the 1D theory the gap in the flux leads to a continued growth of LWA, but the 2D case achieves a quasi-steady state because the decrease in the zonal flux is compensated by the meridional flux produced by the deflected waves. The sideways shedding of PV filaments in Fig. 8 is similar to the findings by Swanson et al. (1997). Farther downstream the PV contours are more confined meridionally, similar to the weak forcing case in Fig. 6.

Fig. 8.
Fig. 8.

As in Fig. 6, but with a stronger wave forcing . Equation (29) is violated first at (see Fig. 9), where the magnitude of meridional displacement grows .

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

Fig. 9.
Fig. 9.

As in Fig. 7, but with a stronger wave forcing , corresponding to Fig. 8. By the zonal flow decelerates to less than half of its initial values in .

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

The separatrices of play a central role in shaping the asymptotic wave envelope. As the waves modify the flow the two separatrices in the upstream region become less separated from each other. Consequently the wavy PV contours begin to cross them and their lobes get compressed along the separatrices toward the stagnation points. Therefore, these upstream separatrices behave like “stable manifolds.” Eventually the part of the upstream PV contours that resides outside the stable manifolds gets deflected backward from the stagnation points along the other set of separatrices and exits the domain through the side boundaries. This other set of separatrices makes up “unstable manifolds” and effectively becomes critical lines for the transient waves: the normal component of their group velocity vanishes there and the PV contours spread along, but do not cross, them. Since the structure of the manifolds changes only modestly during the experiment, the degree of sideways deflection of LWA may be predicted from the amount of PV outside the stable manifolds in the upstream [see a related discussion in Swanson et al. (1997)].

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 (Fig. 9; ). Overall, the flow does not resemble a typical atmospheric blocking in which a wave train is split around a stationary high and/or a cutoff low (Rex 1950).

Now we introduce a cross-stream thickness variation by letting , , and . The choice of is roughly consistent with the values found in Table 1:it corresponds to 60% in the last column. Although the forcing amplitude is smaller than the previous case, the enhanced PV gradient maintains the incident LWA flux comparable; thus, we still refer to this case as “strong forcing.” Neither the dispersion relation nor the threshold definition [(29)] is affected by this modification, but a much larger PV jump allows (29) to be violated at a smaller value of . The results are depicted in Figs. 10 and 11. As the wavefront passes through the diffluent region, the zonal wavenumber of PV anomalies increases as before (Fig. 10; ). However, compared to the previous case the waves are more efficient at decelerating the zonal flow along the jet axis: (29) is violated around first, then drops to zero at (Fig. 11; t* = 4–8). The flow slows down at to the point that it moves faster on the flanks, creating a forward tilt in the PV filaments (Fig. 10; ). The forward tilt of the PV filaments sustains a meridional eddy momentum flux away from the jet axis, since for the phase-averaged eddy momentum flux is , which is positive for where (a northeast–southwest tilt) and negative for where (a northwest–southeast tilt). As a result, the contours of streamfunction begin to spread out in the diffluent region. Meanwhile, the folded PV filaments become unstable in the region of weak shear and start to roll up, and eventually a pair of counterrotating vortices emerges in streamfunction—an anticyclone to the north and a cyclone to the south—reminiscent of a modon (McWilliams 1980; Butchart et al. 1989) and atmospheric blocking described by Rex (1950) (Fig. 10; 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; ). The anticyclone (cyclone) reinforces its circulation by preferentially entraining fluid elements with negative (positive) PV anomalies originated from negative (positive) . A similar selective absorption mechanism has been proposed by Yamazaki and Itoh (2009) as a maintenance mechanism for blocking.

Fig. 10.
Fig. 10.

As in Fig. 6, but for , , and . (left) Note that the value of PV in the green and blue areas are now q* = 2.5 and −2.5, respectively, and the drawn PV contours are . (top)–(bottom) . See text for details.

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

Fig. 11.
Fig. 11.

As in Fig. 7, but for , , and , corresponding to Fig. 10. The actual zonal flow decreases below one-half of initial at shortly after and drops below zero at shortly after . See text for details.

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 (Fig. 11; 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 in and in , since about half the wavelength is replaced by PV originating from the other side of jet axis, whose value differs from the basic state by (Fig. 1). In barotropic dynamics, this net PV anomaly is 100% converted to a net vorticity anomaly. Thus, if in the basic state is entirely due to discontinuity in vorticity (i.e., shear of the cusp jet), the induced vorticity anomaly is comparable in magnitude but opposite in sign with this shear since the latter is [see (41)]. However, when topography augments , the induced vorticity anomaly can overcome the background shear. It not only resists the straining by the background shear but is capable of reversing the shear of the flow through the local eddy–zonal flow interaction. This allows anticyclonic (cyclonic) circulation to form to the north (south) of the jet axis, leading to a block formation. We will render this statement more quantitative shortly.

d. Parameter sweep for and

We have repeated the experiments for a range of forcing amplitude and thickness variation with a reduced (one quarter) model resolution for computational efficiency. The low-resolution model significantly smoothes the PV structure and reduces the magnitude of LWA due to numerical diffusion, which affects the location and timing of block formation but not the emerging structure of streamfunction because the small-scale PV is filtered by the inverse Laplacian operator even at a higher resolution. Table 2 summarizes the results. The minus sign indicates that no blocking occurs. In this case the zonal flow never drops below one-half of the initial value along and the wave train is transmitted downstream apart from dissipation by mixing. (This corresponds to the case a above.) The plus sign denotes partial blocking, in which the zonal flow does drop below one-half of the initial value somewhere along and wave train undergoes significant transformation but no reversal of the zonal wind occurs (as in the case b above). The letter B denotes blocking, in which there is a reversal of the zonal wind along and an anticyclone–cyclone pair emerges in streamfunction (as in case c).

Table 2.

Model behavior as a function of (columns) and (rows). A minus sign indicates no blocking; that is, the zonal flow remains above one-half of the initial value everywhere. A plus sign indicates partial blocking; that is, the zonal flow drops below one-half of the initial value somewhere along , but no wind reversal occurs. The letter B indicates blocking; that is, the zonal flow is reversed somewhere along , and an anticyclone–cyclone pair forms in streamfunction. Each experiment was run up to .

Table 2.

For each value of there is a finite value of that marks the boundary between no blocking and partial blocking. This transition is predictable from the 1D theory [(29)]. As increases the threshold amplitude becomes smaller, because the enhanced PV jump allows the same that satisfies (29) to be produced with a smaller forcing amplitude. It is notable that block formation (B) occurs only for . A heuristic argument for this is as follows. Assuming that a blocking high/cut-off low forms in a circular area with the radius by filling the net vorticity anomaly of magnitude in it, then it induces circulation about along its circumference relative to the initial condition. For this induced circulation to overcome the circulation of the initial zonal flow and reverses its direction along , it is required that
e56
By substituting (50) and , (56) becomes
e57
This sets the lower bound for the block-producing , consistent with Table 2.

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.

Fig. 12.
Fig. 12.

As in Fig. 10, but for instead of 15.4.

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

HN16 defines local wave activity (LWA) as the meridional displacement of PV from zonal symmetry at a given longitude and time. On the Cartesian plane
ea1
where
ea2
is the departure of PV relative to the value of a zonally symmetric, steady reference state at latitude y, and defines the meridional location of the wavy PV contour that carries the value (it can be multivalued; see Fig. 1 of HN16). In the single PV-front problem, only one PV contour can be defined meaningfully and it is initially located at . Thus, from (1),
ea3
At some x where the contour is displaced to a positive , PV is in the segment . Therefore . Substituting in (A1),
ea4
Where the contour is displaced to a negative , PV is in the segment . Therefore . Substituting in (A1) and taking care of the direction of the integral,
ea5
From (A4) and (A5),
ea6
where the reference to is omitted. Hence for the current problem LWA is a sign-definite first-order quantity in the eddy amplitude. If is locally a sinusoidal function , where the constant μ represents phase shift, the phase average of is . Therefore the phase average of is
ea7
which is (19).
To derive the governing equation for in the small-amplitude limit, we start from the linearized mass continuity equation in the barotropic contour dynamics [e.g., Swanson et al. 1997, their (A.6)]:
ea8
which is equivalent to (2) after substituting (4), except that we linearize about the zonal flow that is already modified by the waves. [See also (34).] By multiplying the above equation by , one obtains
ea9
Further multiplying by and using (A6),
ea10
This is analogous to (20) of HN16 (in the barotropic limit): the second term on the left-hand side represents the divergence of the zonal advective flux of LWA, whereas the last term represents the meridional PV (vorticity) flux across . By taking the phase average of (A10) the fast variation in drops out and the first line of (20) is obtained.
As an alternative form of (A10), consider
ea11
where the first identity uses (17) [which also applies to a time-dependent WKB solution] and the last identity uses (A6). Thus, (A10) becomes
ea12
where (12) was used except that was replaced by . The phase average of (A12) results in the last line of (20).

b. Local zonal flow in a barotropic PV front

To see the relationship between LWA and local zonal flow, we need the zonal momentum equation. The barotropic zonal momentum equation on the f plane reads
ea13
where p is dynamic pressure and is a constant density. Velocities are related to streamfunction ψ through and is relative vorticity. To apply (A13) to the present PV front problem at , we define the flow as
ea14
ea15
ea16
ea17
ea18
ea19
ea20
ea21
Here and define a steady basic state that varies slowly in x . It is assumed that P and are an exact solution of (A13). Since is only slowly varying in x, the corresponding meridional velocity V is formally compared to U, but in our problem we choose a flow in which is constant at so we have exactly . The terms and are the slowly varying, time-dependent part of the flow induced by the small-amplitude waves. Their magnitudes are assumed to be much smaller than those of P and . It thus follows that . Finally, and represent the wavy component with fast phase oscillation but a slowly modulated amplitude. Note that does not appear above because the y derivative of is not defined at [there is a jump in across as in (2)]. If the magnitude of does not exceed that of , then is much smaller than υ′ since does not involve fast oscillation within the phase. Similarly, the x derivative of is much smaller than that of . Finally, (potential) vorticity at is governed by the displacement of the PV contour: where the displacement is positive PV is and where negative it is [see (1)].
By substituting the above definitions in (A13), subtracting the basic-state solution and neglecting the small terms, one obtains
ea22
In the above, and we assumed . After taking the phase average to remove fast oscillation in , one obtains
ea23
This is (21).

REFERENCES

  • Altenhoff, A. M., O. Martius, M. Croci-Maspoli, C. Schwierz, and H. C. Davies, 2008: Linkage of atmospheric blocks and synoptic-scale Rossby waves: A climatological analysis. Tellus, 60A, 1053–1063, doi:10.1111/j.1600-0870.2008.00354.x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Anderson, J., 1995: A simulation of atmospheric blocking with a forced barotropic model. J. Atmos. Sci., 52, 2593–2608, doi:10.1175/1520-0469(1995)052<2593:ASOABW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Barnes, E. A., E. Dunn-Sigouin, G. Masato, and T. Woollings, 2014: Exploring recent trends in Northern Hemisphere blocking. Geophys. Res. Lett., 41, 638–644, doi:10.1002/2013GL058745.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Barriopedro, D., R. Garcia-Herrera, A. Lupo, and E. Hernandez, 2006: A climatology of Northern Hemisphere blocking. J. Climate, 19, 1042–1063, doi:10.1175/JCLI3678.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Berggren, R., B. Bolin, and C.-G. Rossby, 1949: An aerological study of zonal motion, its perturbations and break-down. Tellus, 1, 14–37, doi:10.3402/tellusa.v1i2.8501.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bretherton, F. P., and C. J. R. Garrett, 1968: Wavetrains in ingomogeneous moving media. Proc. Roy. Soc. London, 302A, 529–554, doi:10.1098/rspa.1968.0034.

    • Search Google Scholar
    • Export Citation

  • Butchart, N., K. Haines, and J. Marshall, 1989: A theoretical and diagnostic study of solitary waves and atmospheric blocking. J. Atmos. Sci., 46, 2063–2078, doi:10.1175/1520-0469(1989)046<2063:ATADSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Cash, B., and S. Lee, 2000: Dynamical processes of block evolution. J. Atmos. Sci., 57, 3202–3218, doi:10.1175/1520-0469(2000)057<3202:DPOBE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Charney, J., and P. Drazin, 1961: Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66, 83–109, doi:10.1029/JZ066i001p00083.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colucci, S., 1985: Explosive cyclogenesis and large-scale circulation changes: Implications for atmospheric blocking. J. Atmos. Sci., 42, 2701–2717, doi:10.1175/1520-0469(1985)042<2701:ECALSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colucci, S., 2001: Planetary-scale preconditioning for the onset of blocking. J. Atmos. Sci., 58, 933–942, doi:10.1175/1520-0469(2001)058<0933:PSPFTO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Croci-Maspoli, M., C. Schwierz, and H. Davies, 2007: A multifaceted climatology of atmospheric blocking and its recent linear trend. J. Climate, 20, 633–649, doi:10.1175/JCLI4029.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dee, D., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553–597, doi:10.1002/qj.828.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dritschel, D., and M. Ambaum, 1997: A contour-advective semi-Lagrangian numerical algorithm. Quart. J. Roy. Meteor. Soc., 123, 1097–1130, doi:10.1002/qj.49712354015.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams–Bathforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, doi:10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Fyfe, J., and I. Held, 1990: The two-fifths and one-fifth rules for Rossby wave breaking in the WKB limit. J. Atmos. Sci., 47, 697–705, doi:10.1175/1520-0469(1990)047<0697:TTFAOF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Green, J., 1977: The weather during July 1976: Some dynamical considerations of the drought. Weather, 32, 120–128, doi:10.1002/j.1477-8696.1977.tb04532.x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Haines, K., and J. Marshall, 1987: Eddy-forced coherent structures as a prototype of atmospheric blocking. Quart. J. Roy. Meteor. Soc., 113, 681–704, doi:10.1002/qj.49711347613.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Harvey, B., J. Methven, and M. Ambaum, 2016: Rossby wave propagation on potential vorticity fronts with finite width. J. Fluid Mech., 794, 775–797, doi:10.1017/jfm.2016.180.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hoskins, B., M. McIntyre, and A. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877–946, doi:10.1002/qj.49711147002.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Huang, C. S. Y., and N. Nakamura, 2016: Local finite-amplitude wave activity as a diagnostic of anomalous weather event. J. Atmos. Sci., 73, 211–229, doi:10.1175/JAS-D-15-0194.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Huang, C. S. Y., and N. Nakamura, 2017: Local wave activity budgets of the wintertime Northern Hemisphere: Implication for the Pacific and Atlantic storm tracks. Geophys. Res. Lett., doi:10.1002/2017GL073760, in press.

    • Search Google Scholar
    • Export Citation

  • LeVeque, R., 2002: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 580 pp.

  • Lighthill, M., and G. Whitham, 1955: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London, 229A, 317–345, doi:10.1098/rspa.1955.0089.

    • Search Google Scholar
    • Export Citation

  • Luo, D., 2005: Barotropic envelope Rossby soliton model for block-eddy interaction. Part I: Effect of topography. J. Atmos. Sci., 62, 5–21, doi:10.1175/1186.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Masato, G., B. Hoskins, and T. Woollings, 2013: Winter and summer Northern Hemisphere blocking in CMIP5 models. J. Climate, 26, 7044–7059, doi:10.1175/JCLI-D-12-00466.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • McWilliams, J., 1980: An application of equivalent modons to atmospheric blocking. Dyn. Atmos. Oceans, 5, 43–66, doi:10.1016/0377-0265(80)90010-X.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mokhov, I., A. Timazhev, and A. Lupo, 2014: Changes in atmospheric blocking characteristics within Euro-Atlantic region and Northern Hemisphere as a whole in the 21st century from model simulations using RCP anthropogenic scenarios. Global Planet. Change, 122, 265–270, doi:10.1016/j.gloplacha.2014.09.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mullen, S., 1987: Transient eddy forcing of blocking flows. J. Atmos. Sci., 44, 3–22, doi:10.1175/1520-0469(1987)044<0003:TEFOBF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 1993: Momentum flux, flow symmetry, and the nonlinear barotropic governor. J. Atmos. Sci., 50, 2159–2179, doi:10.1175/1520-0469(1993)050<2159:MFFSAT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nakamura, N., and D. Zhu, 2010: Finite-amplitude wave activity and diffusive flux of potential vorticity in eddy–mean flow interaction. J. Atmos. Sci., 67, 2701–2716, doi:10.1175/2010JAS3432.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nakamura, N., and A. Solomon, 2011: Finite-amplitude wave activity and mean flow adjustments in the atmospheric general circulation. Part II: Analysis in the isentropic coordinate. J. Atmos. Sci., 68, 2783–2799, doi:10.1175/2011JAS3685.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Nakamura, H., M. Nakamura, and J. L. Anderson, 1997: The role of high- and low-frequency dynamics in blocking formation. Mon. Wea. Rev., 125, 2074–2093, doi:10.1175/1520-0493(1997)125<2074:TROHAL>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Orlanski, I., 1976: A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys., 21, 251–269, doi:10.1016/0021-9991(76)90023-1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Rex, D., 1950: Blocking action in the middle troposphere and its effect upon regional climate. Tellus, 2, 196–211, doi:10.3402/tellusa.v2i3.8546.

    • Search Google Scholar
    • Export Citation

  • Richards, P. I., 1956: Shock waves on highways. Oper. Res., 4, 42–51.

  • Shutts, G. J., 1983: Propagation of eddies in diffluent jetstreams: Eddy forcing of ‘blocking’ flow fields. Quart. J. Roy. Meteor. Soc., 109, 737–761, doi:10.1002/qj.49710946204.

    • Search Google Scholar
    • Export Citation

  • Swanson, K., 2000: Stationary wave accumulation and the generation of low-frequency variability on zonally varying flows. J. Atmos. Sci., 57, 2262–2280, doi:10.1175/1520-0469(2000)057<2262:SWAATG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Swanson, K., P. Kushner, and I. Held, 1997: Dynamics of barotropic storm tracks. J. Atmos. Sci., 54, 791–810, doi:10.1175/1520-0469(1997)054<0791:DOBST>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Treiber, M., and A. Kesting, 2013: Traffic Flow Dynamics: Data, Models and Simulation. Springer, 506 pp.

  • Tyrlis, E., and B. Hoskins, 2008: Aspects of a Northern Hemisphere blocking climatology. J. Atmos. Sci., 65, 1638–1652, doi:10.1175/2007JAS2337.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, H., and J. Fyfe, 2000: Onset of edge wave breaking in an idealized model of the polar stratospheric vortex. J. Atmos. Sci., 57, 956–966, doi:10.1175/1520-0469(2000)057<0956:OOEWBI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, L., and N. Nakamura, 2016: Covariation of finite-amplitude wave activity and the zonal-mean flow in the midlatitude troposphere. Part II: Eddy forcing spectra and the periodic behavior in the Southern Hemisphere summer. J. Atmos. Sci., 73, 4731–4752, doi:10.1175/JAS-D-16-0091.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yamazaki, A., and H. Itoh, 2009: Selective absorption mechanism for the maintenance of blocking. Geophys. Res. Lett., 36, L05803, doi:10.1029/2008GL036770.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhu, D., and N. Nakamura, 2010: On the representation of Rossby waves on the β-plane by a piecewise uniform potential vorticity distribution. J. Fluid Mech., 664, 397–406, doi:10.1017/S002211201000457X.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Altenhoff, A. M., O. Martius, M. Croci-Maspoli, C. Schwierz, and H. C. Davies, 2008: Linkage of atmospheric blocks and synoptic-scale Rossby waves: A climatological analysis. Tellus, 60A, 1053–1063, doi:10.1111/j.1600-0870.2008.00354.x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Anderson, J., 1995: A simulation of atmospheric blocking with a forced barotropic model. J. Atmos. Sci., 52, 2593–2608, doi:10.1175/1520-0469(1995)052<2593:ASOABW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Barnes, E. A., E. Dunn-Sigouin, G. Masato, and T. Woollings, 2014: Exploring recent trends in Northern Hemisphere blocking. Geophys. Res. Lett., 41, 638–644, doi:10.1002/2013GL058745.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Barriopedro, D., R. Garcia-Herrera, A. Lupo, and E. Hernandez, 2006: A climatology of Northern Hemisphere blocking. J. Climate, 19, 1042–1063, doi:10.1175/JCLI3678.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Berggren, R., B. Bolin, and C.-G. Rossby, 1949: An aerological study of zonal motion, its perturbations and break-down. Tellus, 1, 14–37, doi:10.3402/tellusa.v1i2.8501.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bretherton, F. P., and C. J. R. Garrett, 1968: Wavetrains in ingomogeneous moving media. Proc. Roy. Soc. London, 302A, 529–554, doi:10.1098/rspa.1968.0034.

    • Search Google Scholar
    • Export Citation

  • Butchart, N., K. Haines, and J. Marshall, 1989: A theoretical and diagnostic study of solitary waves and atmospheric blocking. J. Atmos. Sci., 46, 2063–2078, doi:10.1175/1520-0469(1989)046<2063:ATADSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Cash, B., and S. Lee, 2000: Dynamical processes of block evolution. J. Atmos. Sci., 57, 3202–3218, doi:10.1175/1520-0469(2000)057<3202:DPOBE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Charney, J., and P. Drazin, 1961: Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res., 66, 83–109, doi:10.1029/JZ066i001p00083.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colucci, S., 1985: Explosive cyclogenesis and large-scale circulation changes: Implications for atmospheric blocking. J. Atmos. Sci., 42, 2701–2717, doi:10.1175/1520-0469(1985)042<2701:ECALSC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Colucci, S., 2001: Planetary-scale preconditioning for the onset of blocking. J. Atmos. Sci., 58, 933–942, doi:10.1175/1520-0469(2001)058<0933:PSPFTO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Croci-Maspoli, M., C. Schwierz, and H. Davies, 2007: A multifaceted climatology of atmospheric blocking and its recent linear trend. J. Climate, 20, 633–649, doi:10.1175/JCLI4029.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dee, D., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553–597, doi:10.1002/qj.828.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Dritschel, D., and M. Ambaum, 1997: A contour-advective semi-Lagrangian numerical algorithm. Quart. J. Roy. Meteor. Soc., 123, 1097–1130, doi:10.1002/qj.49712354015.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams–Bathforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, doi:10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation