a. Reference state

Since large-scale dynamics may be most succinctly examined from the vantage point of PV, we choose to prescribe a reference state in terms of its PV field q instead of its velocity field. Each of (q_i − ε⁻¹ − βy) and the corresponding ψ_i may be regarded as a sum of two components without loss of generality. Thus, we set q_i − ε⁻¹ − βy = q̂_i + q̃_i and ψ_i = ψ̂_i + ψ̃_i. The zonal mean component of the reference state is denoted by a “hat,” and the zonally varying component by a “tilde.” Denoting a constant zonal mean component in the zonal velocity of the reference state at the two levels by Û_i, we specifically set q̂₁ = −q̂₂ = λ²(Û₁ − Û₂)y ≡ λ²Û_Ty. Here, Û_T is a parameter for global baroclinicity. It follows that ψ̂₁ = −Û₁y and ψ̂₂ = −Û₂y. It is relevant to consider Û₁ > 0 and Û₂ = 0, for the extratropics in winter.

The zonal inhomogeneity of a forcing is introduced solely through q̃_i. We set q̃₂ = 0 and introduce an idealized but plausible structure for q̃₁(x, y) for simplicity as follows:

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where a, b, Δ, y₀, are constants. Fig. 4 shows the reference PV field for a = 1.5, b = 1.5, and y₀ = b + 0.1. The constant value of ε⁻¹ = 10 is not included in these plots. The corresponding ψ̃_i are computed on the basis of q̃_i according to (5). The parameters in q̃₁(x, y) are used to control the spatial structure of the velocity field of a localized jet in the reference state.

When the upper-level PV of a particular reference flow pattern in our model has a stronger local gradient, say by increasing the strength of the PV jump Δ, the horizontal shear in the corresponding flow field would be greater. This relation can be directly verified from the corresponding velocity field determined by PV inversion. Our model setting serves to mimic a simultaneous increase in the PV gradient and horizontal shear of an observed jet by increasing Δ. Therefore, an effect due to an increase in the PV gradient in this particular setting can be interpreted as an effect due to a corresponding increase in the barotropic shear. This is always true when we compare the gradient of a specific PV field with that of another PV field as long as they have the same pattern. However, we cannot make the same inference if two PV distributions with different patterns are compared. This is an important point to keep in mind in order to avoid unnecessary confusion.

The two key parameters that strongly affect the instability in this model are Û_T and Δ. Since Û_T is a model counterpart of the global measure of baroclinicity, it should increase by about 10% from early to midwinter. One must take special care in interpreting the implication of a particular value of Δ in relation to a winter mean flow. By PV inversion, we determine the velocity field of a reference state for given Û_T and Δ. A meaningful value of Δ should be prescribed on the basis that its corresponding velocity field is compatible with the observed time mean wind field. In particular, we quantify the degree of zonal variation in the basic baroclinicity by

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where U_i is the zonal velocity of the reference state. If we use (Û_T, Δ) = (2.2, 5.5) as a midwinter condition, we get (U₁ − U₂)_max = 4.5 and (U₁ − U₂)_min = 2.4. The corresponding value of γ is equal to 0.47. If we use (Û_T, Δ) = (2., 2.5) as an early winter condition, we get γ = 0.35 because the corresponding values for (U₁ − U₂)_max and (U₁ − U₂)_min are 3.1 and 2.0, respectively. Therefore, the zonal asymmetry of baroclinicity used in such a model analysis only increases by (0.47 − 0.35)/0.35 = 34%. It should be emphasized that more than doubling the value of Δ only implies an increase of 34% in the zonal asymmetry of baroclinicity. Such a change is compatible with the counterpart observed change noted in the introduction. It is therefore justifiable to use (Û_T, Δ) = (2., 2.5) and (Û_T, Δ) = (2.2, 5.5) in this study as plausible early and midwinter conditions, respectively, for the purpose of investigating the dynamics of MWM.

The flow of the reference state is indeed a localized baroclinic jet much stronger in the upper level under the model midwinter condition, Û_T = 2.2 and Δ = 5.5 (Fig. 5). The localized jet is about 7000 km long. It has a meaningful maximum speed of 73.7 m s⁻¹ in the upper level and a maximum speed of 27.4 m s⁻¹ in the lower level. The jet is centered at x = ±12 and has a downstream diffluence/confluence region between the two waveguides at y = ±G. These characteristics are compatible with those of the observed Pacific jet.

a. Reduction of growth rate

The instantaneous growth rate is defined by

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where 〈E〉 = ∫∫_x,y[∑_i=1,2 ½(u²_i + v²_i) + 2λ²θ²]dxdy is the domain-integrated energy of the disturbance. Let us begin by examining how the growth rate would vary with Δ for a fixed value of Û_T. The evolution of the instantaneous growth rate for Δ = 0, 2.5, 4.5, and 5.5 with a fixed value of Û_T = 2.0 are shown in Fig. 6. The asymptotic growth rate is largest for the case of a horizontally uniform baroclinic shear flow (Û_T = 2., Δ = 0). It is ∼0.68, corresponding to an e-folding time of ∼1.5 days. For a localized baroclinic jet with Δ = 2.5, the asymptotic growth rate lowers to ∼0.39, corresponding to an e-folding time of ∼2.5 days. For a more localized basic baroclinic jet characterized by Δ = 4.5 and 5.5, the growth rate decreases to 0.26 and 0.2, respectively. As an unstable disturbance propagates through a more localized baroclinic jet, its structure varies more substantially. The rate of extracting energy from the basic flow correspondingly fluctuates more in time. This is why the asymptotic growth rate for the case of Δ = 5.5 oscillates in time.

As noted earlier, Û_T = 2. and Δ = 2.5 may be taken to be the relevant values for early winter (November), and Û_T = 2.2 and Δ = 5.5 for midwinter (January). For Û_T = 2.2 and Δ = 5.5, the growth rate is found to be 0.28. Hence, the growth rate is reduced by (0.39 − 0.28)/0.39 = 28% from the model early winter condition to the model midwinter condition. Such result is consistent with a reduction of the intensity of the storm track. In summary, the linear instability analysis confirms that the growth rate under plausible model midwinter condition is indeed significantly smaller than that under plausible early winter condition.

b. Structure and local energetics

Additional insight into the nature of the instability for a midwinter condition may be gained from an examination of its structure and local energetics (generation and redistribution processes). A snapshot of the unstable perturbation streamfunctions ψ₁ and ψ₂ at large time (effectively the most unstable normal mode) is shown in Fig. 7. Variable ψ_i is normalized to maximum unit magnitude at the upper level. The dominant wavelength is ∼5 000 km. The most intense wave packets are found slightly downstream of the jet core, in the region −12 ≤ x ≤ −6. The disturbance in the upper level is about 3 times stronger than that in the lower level. There is a distinct westward vertical tilt. The horizontal tilt at both levels is generally in the direction of the basic horizontal shear in the jet exit region (i.e., not leaning against the shear). These tilts have clear implications on the local energetics of the disturbance.

The distributions of the rms of ψ₁ and ψ₂ of the most unstable normal mode can be used as an estimate of a model storm track according to linear dynamics. Since only the pattern of ψ₁ is unique, we normalize the rms values to a maximum of unity at the upper level. Figure 8 shows the result of ψ_i as calculated from the streamfunction field from t = 180 to 240 in our model integration. The result shows a highly localized model storm track at the upper level with its center being slightly downstream of the background jet. The synoptic variability is much weaker in the diffluent/confluent region of the background flow. The model storm track at the lower level is also located downstream of the jet. The intensity of the lower-level storm track is about 35% of that of the upper level.

The vertically integrated local perturbation energy equations for this model can be written as follows (Cai and Mak 1990):

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where {K} = ∑_i=1,2 K_i = ∑_i=1,2 ½(u²_i + υ²_i) is the total local perturbation kinetic energy with v = (u_i,υ_i) being the perturbation velocity; {P}= 2λ²θ² is the local perturbation potential energy with θ = (ψ₁ − ψ₂)/2. Here, {−V · ∇K} = −∑_i=1,2V_i · ∇K_i and {−V · ∇P} = −(V₁ + V₂)/2∇{P} are the advection of perturbation kinetic energy and potential energy by the basic flow, respectively. Also, V_i is the basic velocity, v_d is the irrotational part of the ageostrophic wind, and φ_a is the ageostrophic part of the geopotential. Then {−∇ · (φ_av + φv_d)} = −∑_i=1,2∇ · (φ_aiv_i + φ_iv_di) is the convergence of energy flux associated with the ageostrophic component of the flow and {E · D} = ∑_i=1,2 E_i · D_i is the local generation rate of kinetic energy to be calculated as a scalar product of two pseudovectors. Vector E_i = [½(υ_i − u_i), − u_i υ_i] is a measure of the local structure of the disturbance at each level, and vector D_i = (∂U_i/∂x − ∂V_i/∂y, ∂V_i/∂x + ∂U_i/∂y) is a measure of the stretching and shearing deformation of the basic flow at each level. The vertical velocity at the midlevel is w = −ω. The term {F · T} represents the net generation rate of perturbation potential energy. It has two parts, {F · T} = {F_H · T_H} + {F₍₃₎T₍₃₎}, where {F_H · T_H} = −4λ²θJ(ψ,Θ), and {F₍₃₎T₍₃₎} = −4ε⁻¹wθ with ψ = (ψ₁ + ψ₂)/2, Θ = (ψ₁ − ψ₂)/2. The former represents a conversion rate from basic potential energy to perturbation potential energy. The latter represents a conversion rate from perturbation kinetic energy to perturbation potential energy. The local conversion from perturbation potential energy to kinetic energy is therefore C(P′, K′) ≡ − {F₍₃₎T₍₃₎}. A normalized perturbation with unit domain-averaged perturbation energy will be used in the computations of its local energetics.

Figure 9a shows that C(P′, K′) is positive almost everywhere and has a narrow band of particularly large values distinctly downstream of the jet core. The domain-integrated value of 〈C(P′, K′)〉 is 119.6. The baroclinic energy conversion is associated with the westward tilt of the wave packets at the upper level relative to those at the lower level.

A wave packet also tends to be meridionally strained as it approaches the diffluent part of the basic flow. This process weakens a wave packet and even splits it into three separate wave packets as seen in Fig. 7a. Figure 9b shows that nevertheless there is some local barotropic conversion from the basic flow to the disturbance in a narrow zone near y = 0 particularly in the region upstream of the jet. The related positive values of E · D contribute to local elongation of the unstable wave packets. The domain-integrated value of 〈E · D〉 is, however, negative, −49.6. This value is significant because |〈E · D〉|/〈C(P′. K′)〉 = 41%. As discussed in the introduction, the nature of instability of a localized jet with such a characteristic may be interpreted as a generalized barotropic governor effect. This stabilization influence is the consequence of an increase in the local deformation of the basic flow from early to midwinter in spite of a simultaneous enhancement of the local baroclinicity.

c. Linear instability regimes

In order to get a broad perspective of the instability of a localized baroclinic jet, we repeat the instability calculations for different combinations of (Û_T, Δ) in a substantial range of the two key parameters. The ratio of the two fundamental energy conversion rates, α ≡ 〈E · D〉/〈C(P′, K′)〉, is a meaningful indicator of the physical nature of the instability. For instance, along the Û_T axis of Fig. 10 (Δ = 0), we have α = 0 because barotropic process is absent by definition. Three instability regimes are identified when we use the sign of α to broadly distinguish the nature of the instability (Fig. 10). The instability regime relevant to our hypothesis for MWM is characterized by α < 0 with 〈C(P′, K′)〉 > 0 and 〈E · D〉 < 0. It is referred to as regime I.

The changes in the structure of the time mean flow from November to January generally differ from year to year. The model counterpart of such possible changes are indicated by arrows with labels A, B, or C in Fig. 10. The MWM in a particular year would be expected to be less pronounced if the transition is close to arrow C; whereas it would be more pronounced if the transition is close to arrow A. Arrow B is representative for a year with an average MWM. Calculations confirm that a combination of a relatively small value of Û_T and a relatively large value of Δ in midwinter would be a favorable condition for MWM, and vice versa.

If we move along a path of decreasing Û_T and increasing Δ in Fig. 10, we would deal with a more and more zonally localized basic flow. The barotropic process would play a more and more positive role. The instability would change progressively from being primarily baroclinic to primarily barotropic in character. We refer to the instability solely attributable to the barotropic process as regime III. The regime in which both baroclinic and barotropic processes contribute positively to the instability is referred to as regime II. A more elaborated discussion of the nature of regimes II and III is given in the appendix.

a. Nonlinear model storm track

We next test our hypothesis for MWM in the context of nonlinear dynamics. Our computations use (Û_T = 2., Δ = 2.5) and (Û_T = 2.2, Δ = 5.5) as plausible settings for the forcing in early winter and midwinter, respectively. Noting that (6c) is a boundary condition for the zonal mean part of ψ_i, we separately determine the zonal mean part and the wave part of ψ_i for given q_i fields. Denoting them by square bracket and asterisk, respectively, we define

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The wave part ψ*_I can be solved straightforwardly from the known q*_i since they are related by

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with boundary conditions

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Taking the y derivative and the zonal mean of (5), we get the equations that relate the zonal mean part of the zonal velocity [u_i] = −∂[ψ_i]/∂y to the zonal mean part of q_i:

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Equation (17) with condition (18) can be then solved numerically in a gridpoint representation.

We numerically integrate Eq. (4) for about 450 days in each case, whereby robust statistical properties of the flow can be established. The damping parameters are τ = 30 and κ = 4.15 × 10⁻⁴, which correspond to 1-h damping time for the smallest scale. The numerical scheme has an additional weak damping effect. A normalized unstable normal mode of the corresponding reference state is used as the initial disturbance. The maximum velocity of the initial disturbance is chosen to be 5 m s⁻¹. This disturbance quickly equilibrates by the nonlinear feedback and damping process. The model output is archived twice a day and the evolution in the last 300 days is used to diagnose its statistical properties.

The bandpass filter in Blackmon (1976) is used to isolate the fluctuations at each grid point that has time scales primarily between 2.5 and 6 days. The root-mean-square of the bandpass ψ_i (rms{ψ_i }_B for short) shows that the model produces and maintains a well-defined model storm track under early winter condition, Û_T = 2., Δ = 2.5 (Fig. 11). The maximum intensity of rms{ψ₁}_B is slightly more than twice as strong as rms{ψ₂}_B, 0.62 versus 0.27. These intensities correspond to about 62 and 27 m in upper- and lower-level geopotential heights respectively. This model storm track is located distinctly downstream of the reference baroclinic jet. It is encouraging to find such qualitative similarity between this model storm track and observation.

The model storm track under midwinter condition, Û_T = 2.2, Δ = 5.5, is shown in Fig. 12. The intensity in each level is considerably weaker than the counterpart under early winter conditions. The maximum intensity of rms{ψ₁ }_B is about 3 times larger than that of rms{ψ₂}_B, 0.40 versus 0.15. There is a shorter downstream extension of the storm track relative to the reference baroclinic jet under midwinter conditions. It is distinctly weaker than the early model winter storm track by (0.62 − 0.44)/0.62 = 29%. This model result is relevant to the observed reduction of the Pacific storm track.

It is also instructive to compare the linear model storm track with the nonlinear model storm track (Fig. 12 and Fig. 8). They are fairly similar to one another. The nonlinear storm track at the upper level has a smoother structure in the diffluent/confluent region than that of the linear storm track, as expected. The nonlinear storm track at the lower level is also more localized than the linear counterpart. The results of both linear and nonlinear analyses therefore support the hypothesis that MWM is partly a consequence of enhanced negative impact upon the eddies by the barotropic process associated with the deformation field of the Pacific jet in spite of an increase in its local baroclinicity from early to midwinter.

b. Model time mean flow and nonlinear feedback of the eddies

Recall that we have used the observed monthly mean flows as a guide for prescribing the two key forcing parameters (Û_T, Δ) of the reference state for early and midwinter. It is therefore necessary to check a posteriori whether the nonlinear model time mean flow has reasonable resemblance to the corresponding reference flow. If not, the forcing condition cannot be regarded as being appropriate for that time of year. Figure 13 shows the model time mean wind field under the midwinter forcing condition. The time mean flow in the upper level is only slightly different from Fig. 5a. The time mean flow in the lower level is also quite similar to Fig. 5b, although it is noticeably stronger. It means that the background baroclinicity has been generally reduced and the barotropic component of the flow has been significantly strengthened because of the feedback effects of the eddies. The same is also true for the early winter case. These results provide reassurance that the difference in the model forcing between early and midwinter is relevant to the corresponding difference in the atmosphere.

We highlight the feedback effects of the eddies by calculating

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where u_i are the zonal velocities of the departure flow component. The results for the midwinter condition are shown in Fig. 14. These values are small compared to the counterparts of the reference state. It is confirmed that there is an overall reduction in the baroclinicity since the domain-average value in Fig. 14a is negative. It is of interest to find a small increase in the downstream region of the background jet. Figure 14b reveals that the barotropic feedback is much stronger. The positive values in the central region together with the negative values on both sides would enhance the meridional shear particularly in the downstream region of the jet.

The feedback effect of the eddies on the background flow can be most succinctly diagnosed from the perspective of PV. Figure 15 shows the time mean departure component of the PV field at the upper and lower levels for midwinter condition (Û_T, Δ) = (2.2, 5.5). These values are such that the net PV gradient along the two background waveguides is reduced. The largest modification occurs in the central region of the domain between the two waveguides at the upper level of the reference state. For example, the PV is increased (decreased) in the northern (southern) portion of this region. Furthermore, the PV in the northern (southern) half of the domain at the lower level is increased (decreased). In other words, the eddies statistically transport PV in the down-gradient direction as they must, giving rise to a total flow with weaker PV gradient. We may infer that the flow is stabilized to the greatest possible extent by eddies in the equilibrated state. Since the instability is stronger under early winter conditions, the feedback effects are found to be correspondingly stronger.