1. Introduction

Localized jets are ubiquitous in the atmosphere and the winter mean large-scale localized jets in the Northern Hemisphere are well-documented (e.g., Fig. 6.2 in Holton 1992). There have been a number of analyses of the instability properties of such general shear flows in idealized settings (e.g., Frederiksen 1979; Pierrehumbert 1984, hereafter P84; Mak and Cai 1989; Cai and Mak 1990; Farrell 1989; Samelson and Pedlosky 1990; Borges and Hartmann 1992; and others), as well as in realistic settings (e.g., Frederiksen 1983; Simmons et al. 1983; Lee and Mak 1995; Branstator and Held 1995; Huang and Robinson 1995; and others). However, the understanding of this general instability problem is not nearly as comprehensive as that of a parallel shear flow. For example, the reasons for the high sensitivity of the instability properties of this class of flows to seemingly minor features of a basic flow are still not clear.

It has been particularly instructive to analyze the instability of a zonally varying geophysical flow with the notions of absolute instability and convective instability in the context of Wentzel–Krames–Brillouin (WKB) approximation (P84; also, see Huerre and Monkewitz 1990, in a more general context). It was found that an unstable mode due to absolute instability has zero group velocity and a highly localized spatial structure. It is referred to as a local mode. On the other hand, an unstable mode that recycles through the streamwise boundaries of a domain is attributable to convective instability. It spans over the whole model domain and is referred to as a global mode. A good approximation for the properties of absolute instability was deduced in P84 from the local properties of the basic flow at the point of maximum baroclinicity (supercriticality). Pierrehumbert further suggests that to have a well-defined local mode, the ratio of maximum baroclinicity to minimum baroclinicity downstream must be sufficiently large, and two successive peaks of maximum baroclinicity must be sufficiently far apart.

It is noteworthy that only disturbances that have no variation in the transverse direction of the jet (i.e., ∂/∂y = 0, with x being the streamwise direction) are considered in P84. This restriction automatically makes two of the terms in the governing potential vorticity equation identically zero, namely u′q_x and υq^′_y where q is potential vorticity, regardless of the particular structure of a localized flow. It enables one to reduce an inseparable eigenvalue–eigenfunction problem into a separable one. But this is not justifiable in general as noted in P84. It follows that the location of maximum supercriticality of a generic localized jet in the WKB sense is not identifiable a priori in general. Nor would an unstable local mode necessarily have zero group velocity. It is doubtful that the properties of a local mode can be deduced from any local property of a basic flow as it was done in P84. The general understanding of the instability of a zonally varying flow without a restriction of ∂/∂y = 0 on a disturbance has yet to be established.

The instability of a parallel flow can be most succinctly interpreted in terms of wave resonance (Bretherton 1966; Baines and Mitsudera 1994; and others). By wave resonance we mean that instability results from mutual reinforcement of two constituent wave components with the same zonal wavelength that are stationary relative to one another. Their relative stationarity arises from the differential Doppler effect of the basic parallel shear flow. Their interaction stems from the process of mutual advection of the basic potential vorticity. This conceptual interpretation of instability is applicable to all parallel flows including barotropic or baroclinic shear flows with or without rotation (e.g., Hoskins et al. 1985; and others) or even to a shear flow in a nonhydrostatic system (Mak 2001). A general form of wave resonance is wave-packet resonance that could give us a better feel for the instability of a zonally varying flow.

The essence of the instability of a localized jet can be most simply ascertained in the context of a barotropic model without losing too much generality since barotropic and baroclinic problems are mathematically homomorphic. The findings would be meaningful as long as the jet under consideration is physically realizable and resembles typical atmospheric jets. Let us then consider a basic flow in the form of a barotropic localized westerly jet. To the north (south) of this localized jet, the relative vorticity adjacent to the jet core must be cyclonic (anticyclonic). An idealized form of such a localized jet has a finite region of uniform cyclonic (anticyclonic) vorticity to the north (south). The remaining part of the domain has zero relative vorticity (ζ = 0). The gradient of basic vorticity along the boundaries of these two regions would be very large, but is zero elsewhere. Hence, the beta effect may not be entirely negligible.

The localized jet under consideration could clearly support wave packets. They would tend to propagate along the boundary of the region of cyclonic (anticyclonic) basic vorticity in a clockwise (anticlockwise) direction. It would be possible for the two wave packets to resonantly interact. We have then reason to hypothesize that this zonally varying flow can become unstable by virtue of wave-packet resonance.

2. Model formulation

At any particular time, if a ψ field and a corresponding ζ field are known, one can integrate (1) to get a ζ field at the next time step. This new ζ field in turn can be used to get a new ψ field by solving

²ψζ,(2)

with the boundary conditions,

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This procedure is repeated for any length of time relevant to linear dynamics. When the basic flow is unstable, the disturbance would evolve towards the most unstable normal mode at large time, regardless of the structure of the initial disturbance.

To relate the model to the extratropical atmosphere, we choose U = 12 m s⁻¹ and L = 3000 km. Then the domain would be 12 000 km long and 6000 km wide. One unit of time would be 2.5 × 10⁵ s, roughly 3 days. The nondimensional value of β would be (β_dimL²/U) = 12. The time integration of (1) is performed using the Euler backward scheme with a time step Δt = 4 × 10⁻³ that corresponds to 18 min. The spatial derivatives in (1) are approximated by center differences. The domain is depicted with 101-by-101 grid points. The dimensional grid distances are Δx = 120 km and Δy = 60 km. Such resolution is amply adequate because essentially the same instability results are obtained with only 51-by-51 grid points.

a. Basic flow

We consider a generic nondivergent localized jet alluded to in section 1. For convenience, we analytically define a zonally asymmetric localized basic vorticity field in terms of two-dimensional step functions:

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where G(x, y) = (xa)² + (yb)². The major simplifying feature is the presence of two closed contours of basic vorticity gradient. They serve as the only waveguides for the movement of wave packets as well as the locations of change in their intensity. One might add that allowing more closed contours in the ζ field would only complicate but not qualitatively change the problem.

The two regions of uniform basic vorticity ζ are half-ellipses displaced relative to one another in the zonal direction. The zonal and meridional scales of ζ are prescribed through the parameters a, b, and r. The intensity and degree of zonal asymmetry of ζ are controlled by the parameters λ and c, respectively. The uniform basic vorticity used in the experiments of this study is set to ±4 × 10⁻⁵ s⁻¹ amounting to λ = 10. The corresponding basic streamfunction ψ is determined by solving the Poisson equation,

²ψζ(5)

subject to the boundary conditions,

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Here, ψ_N is an additional free parameter related to the presence of rigid lateral boundaries. The velocity field is V = [(u = −ψ_y), (υ = ψ_x)]. The fine jaggedness in ζ along each boundary is inconsequential to the corresponding ψ field due to the scale factor associated with the Laplacian operator. It follows from the linearity of Eqs. (5) and (6) that ψ, and hence the zonal velocity component, consists of one part associated with ζ and another part with ψ_N. The domain-averaged zonal velocity is 〈u〉 = −ψ_N/2.

3. Results

b. Energetics

Further insight into the physical nature of the instability can be deduced from a local energetics diagnosis. The equation that describes the rate of change of the local disturbance energy in nondimensional form can be written as (Mak and Cai 1989)

K^′_tEDVKvp⁽¹⁾(8)

Here, K′ = (1/2)(u′² + υ′²) is the local perturbation energy, and E = [(1/2)(υ′² − u′²), (−u′υ′)] is a measure of the local structure of the disturbance. The perturbation pressure is nondimensionalized as p = (p/ρU²)_dim. Parameter p⁽¹⁾ is the ageostrophic part of p (unbalanced part), defined by

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Variable p⁽¹⁾ can be determined by solving the following equation:

²p⁽¹⁾u^′_xU_xu^′_yV_xυ^′_xU_y(9)

subject to boundary conditions

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Equation (8) indicates that K′ changes as a result of three processes, namely, (i) conversion from basic kinetic energy to K′, (ii) advection of K′ by the basic flow, and (iii) convergence of ageostrophic pressure flux (wave-energy flux) associated with the unbalanced part of the pressure. The role of rotation is manifested in the term (iii). If there were no rotation, the ageostrophic pressure would be just the total perturbation pressure.

The domain integral of (i), of an intensifying disturbance, has a positive value. That would be so if the E vector of a perturbation is spatially correlated with the basic D vector (Fig. 1c). The domain integral of (ii) and (iii) are individually zero. They can only play the role of redistributing the perturbation energy in the domain. All three processes contribute to determining the instantaneous shape of a disturbance. We closely examine the structure of the three terms on the right-hand side of (8) for each disturbance under consideration. It should be added that the process (iii), in a counterpart baroclinic setting, is referred to as convergence of “ageostrophic geopotential flux” (Cai and Mak 1990) and is the basis for the notion of “downstream development” in a nonlinear regime (Orlanski and Chang 1993).

Figure 4a shows that the values of the corresponding conversion rate of energy E·D for the unstable normal mode are mostly positive and are highly localized downstream of the jet. Most of the energy conversion takes place in a region of stretching rather than shearing deformation. Figure 4b shows the configuration of −∇·(VK′). This process simply spreads some K′ in its downstream direction. Figure 4c shows the configuration of −∇·(v′p⁽¹⁾). This process of convergence of ageostrophic pressure flux is essentially a process of dispersion. These three terms have the same order of magnitude. The E·D term is the largest of the three. Figure 4d shows that the sum of them gives rise to a pattern of ∂K′/∂t that coincides well with the pattern of ψ′ field. This makes it possible for the disturbance to intensify as a stationary local mode.

4. Influence of the beta effect

In the second experiment we use β = 0 but leave all other parameters unchanged. The influence of the beta effect can be deduced from the difference between the instability properties in this experiment and those in the first experiment. Figure 5 shows pronounced periodic fluctuation in the growth rate of the asymptotic state. This feature is a manifestation of the generic form of an unstable propagating normal mode,

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where ω = ω_r + iω_i is the eigenvalue. Apart from the amplification factor, the structure varies between two patterns at the extreme phases of each period of 2π/ω_i. This normal mode has a nondimensional period of 2π/ω_i ∼ 5.6 (17 days) and a growth rate varying between 0.95 and 1.25. The mean value is ω_r = 1.1 (∼0.44 × 10⁻⁵ s⁻¹). The variation of the structure would naturally result in different degrees of effectiveness for the disturbance to extract energy from the basic flow. The period is related to the time that it takes for a wave packet to complete a trip around the boundary of the half-ellipse region or through the entire domain.

Figure 6 shows the vorticity perturbation field at t = 16.8, 18.8, 20.8, 22.4 within one period. The small positive vorticity anomaly on the western end of the wave packet at t = 16.8 intensifies as it moves eastward. At the end of one period (t = 22.4), it has evolved to a shape like its adjacent positive vorticity anomaly at t = 16.8. The latter may be then thought of as a constituent disturbance originated one period earlier. By t = 22.4, the original vorticity anomaly has evolved to virtually the same shape as the third positive vorticity anomaly back at t = 16.8. Therefore, the third positive vorticity anomaly at t = 16.8 may be also interpreted as a constituent disturbance that was originated two periods earlier. In other words, a snapshot of a disturbance such as t = 16.8 may be thought of as consisting of three generations of constituent disturbances that systematically move eastward while changing their shapes and intensity. In the absence of the beta effect in the second experiment, the constituent waves in the wave packets are able to propagate eastward along the boundaries of the half-ellipses. Although it is a structurally local mode, the instability is attributable to propagating wave-packet resonance. It should be noted that this is an example of convective instability without recycling through the streamwise boundaries of the domain.

The closed contours of |∇ζ| also provide a setting for the wave packet to continually reseed itself without having to recycle through the whole domain. The westernmost positive vorticity anomaly of the wave packet at t = 22.4 may be viewed as a newly selfreseeded constituent disturbance at the entrant region of the jet. The newly reseeded constituent disturbance is more intense than that of the last cycle, thereby leading to an accumulative growth of the wave packet as a whole. The reseeding process in a disturbance becomes more understandable in the context of a disturbance without the constraint of ∂/∂y = 0. The corresponding structures of the perturbation streamfunction field are not shown for brevity.

5. Impact of a domain-averaged zonal velocity component in the basic flow

The third experiment is performed for the purpose of highlighting the influence of a domain-averaged westerly component in a localized westerly jet upon its instability. Here, we use ψ_N = −0.75 and the same values for all other parameters as those in the first experiment. The corresponding 〈u〉 is a modest westerly, 〈u〉 = −ψ_N/2 = 0.375, which corresponds to 4 m s⁻¹. Figure 7 shows that the disturbance has not evolved to its asymptotic state even after 22 units of time (66 days) since the growth rate still has not settled down to a regular oscillation. It suggests that the growth rate of the most unstable mode is only slightly larger than that of the second unstable mode. This has been verified with a modal instability analysis to be reported separately. The mean value of fluctuating growth rate is about 0.7, which is considerably smaller than the mean value in Fig. 5 (1.1). In this case, the intensifying disturbance vigorously recycles through the streamwise boundary (x = ±2). The instability process is therefore due to propagating wave-packet resonance (convective instability). The eastward recycling of the disturbance through the streamwise boundaries stems from the fact that influence of this 〈u〉 is large enough to overcome the beta effect. This feature is clearly evident as we examine the structure of ψ′ at different times, such as t = 18.0 and t = 19.6 (Fig. 8). At t = 18.0, when the growth rate has a minimum value, ψ′ is a wave train most intense in the southern half of the domain further downstream of the jet exit. At t = 19.6, when the growth rate has a maximum value, ψ′ is a wave train most intense in the northern half of the domain and closer to the jet exit. At the intermediate times, the disturbance propagates through the boundary (x = ±2) and broadly spans over the whole domain. It has been verified that when a weaker westerly 〈u〉 is used, the growth rate of the disturbance at large time would fluctuate less.

The fourth experiment is intended to verify an inference readily deducible from the result of the third experiment. Here, we use ψ_N = 0.5, meaning that there is a weak domain-averaged easterly, 〈u〉 = −0.25 (i.e., 3 m s⁻¹). Figure 9 shows that the asymptotic growth rate in this case also regularly fluctuates between 1.1 and 1.5. So this unstable mode is also interpretable as resulting from propagating wave-packet resonance (convective instability). This propagating mode has a period of ∼4.8 (14.4 days). This is confirmed by the fact that the structure of ψ′ at t = 7.6 is identical to that at t = 10, except with an opposite sign. The mode has a mean growth rate of ∼1.3. Unlike the disturbance in the third experiment, this one propagates westward instead of eastward. This can be clearly seen in Fig. 10 showing the structure of ψ′ at t = 10.0 and 10.8. At these times, the asymptotic growth rate has minimum and maximum values, respectively. The larger instantaneous growth rate at t = 10.8 is associated with a closer overall alignment of the E vector field in ψ′ with respect to the D vector field of the basic flow. The easterly 〈u〉 and the beta effect in the fourth experiment both favor westward movement of a disturbance. Their combined effect overcomes the advective influence of the localized westerly jet. As a result, the unstable mode would naturally propagate in the upstream direction.

6. Concluding remarks

This analysis has delineated the nature of instability of a class of zonally varying barotropic flows. The disturbances in this analysis do not have the restriction of ∂/∂y = 0 as those in P84. In order to highlight the basic aspects of the instability mechanism, we deliberately analyze a generic localized jet that has essentially only two closed contours of |∇ζ| in the domain. Such contours serve as waveguides of wave packets and locations of their mutual interaction. They also highlight how disturbances might selfreseed themselves. The model results validate the hypothesis that instability of a localized jet is in essence a process of wave-packet resonance. The notions of steady and propagating wave-packet resonance are akin to the notions of absolute and convective instability. Wave-packet resonance arguably gives us a better feel for the physical nature of the instability mechanism. As parallel shear flows are a special class of zonally varying flows, so is wave resonance a special version of wave-packet resonance. This conceptual connection enables us to relate the seemingly bewildering complexity in the instability of a localized jet to the simplicity in the instability of a parallel flow.

Whether or not the instability of a zonally varying flow is due to steady or propagating wave-packet resonance depends upon the values of domain-averaged zonal flow component 〈u〉 and β. Four experiments illustrate the dynamical reasons for their influences. The substantial differences in the instability properties of this generic jet in the four experiments highlight their sensitivity to the conditions of the background flow. The gradient of the basic vorticity is associated with the second derivatives of the basic velocity field and the domain-averaged zonal flow component is associated with its area integral. Such properties are difficult to estimate by visual inspection of a flow field. Hence, it is not hard to see why the instability properties of a zonally varying flow are generally sensitive to seemingly minor features of a representative basic flow. A number of other experiments using localized jets of different sizes and shapes have also been made. Those results corroborate the conclusions stated above.