a. The relationship between F_1,yy and supercriticality

As stated earlier, the relative contribution of the terms on the rhs of (7) toward F₂ will be evaluated at y = 0, the jet center (see Fig. 1). For the integrations used to calculate the terms displayed in Fig. 1, κ_M is fixed at 0.2, and the value of β is varied. The linear stability boundary is located at β = 0.26. That is, for β ≥ 0.26, an infinitesimal amplitude arbitrary perturbation, imposed on the radiative equilibrium flow, decays in time. However, as shown in Fig. 1a (see the dotted curve, S₁, in Fig. 1a, which displays F₂ directly calculated from the numerical model; also see Fig. 3 in LH), finite-amplitude wave solutions are obtained in the subcritical region (0.26 ≤ β ≤ 0.37) if the initial state is comprised of finite-amplitude waves and a corresponding zonal mean jet flow. In practice, the model was first run with β = 0.20 for 5000 (dimensionless) model days. The day-5000 solution was used as the initial condition for a β = 0.23 calculation, and the model was run for another 5000 days. The solution at the end of this integration was in turn used as the initial condition for a β = 0.26 calculation. The same process was repeated for β = 0.29, 0.32, 0.35, and 0.37. If β is increased to 0.38, starting from the finite-amplitude solution for β = 0.37, the wave field decays to zero and the zonal mean state becomes that of the radiative equilibrium state. The values displayed in Fig. 1 are evaluated using the model output from the last 3000 days for each of the integrations.

In addition to S₁, Fig. 1a displays the sum of the rhs of (7) (denoted by S₂), and the sum of the rhs of (7) excluding the diffusion term (denoted by S₃). Because the semi-implicit scheme is used to calculate the damping terms, the terms S₁ and S₂ are very similar, but not identical. Figure 1a shows that S₁ and S₂ are essentially identical. More importantly, because the contribution by the D_z term is less than 5% (cf. S₃ with S₂), we focus on the first two terms on the rhs of (7).

As the supercriticality decreases (i.e., as β increases), the contribution by the term (hereafter called term I) toward F₂ changes from negative to positive (see the thin solid line with circles in Fig. 1b). Therefore, a positive value for this term is not a necessary condition for the occurrence of subcritical instability because this term is still negative for β = 0.26 and 0.29, a subcritical part of the parameter space. If term I is positive, as (3) states (note that F_1,yy = −l₁²F₁), > Û_e (solid line with triangles), as long as the diffusion term is negligible. The sign of term I in Fig. 1b implies that only part of the subcritical region coincides with a self-maintaining jet. However, this does not result in the zonal mean flow being more supercritical than the radiative equilibrium state, since, as Fig. 1b shows, [(∂Q₂/∂y) − (∂Q_e2/∂y)] (hereafter called term II) is always positive. This is because the effect of term I, which is proportional to − Û_e, is offset by the influence of the horizontal curvature term, −U_2,yy (dotted curve with circles). Thus, in the time mean, term I is not strong enough to drive the zonal mean state toward being more supercritical than the radiative equilibrium state; that is, (∂Q₂/∂y) < (∂Q_e,2/∂y). However, as we will see in the next section, the process represented by term I plays an important role in the O(1) wave equilibration.

b. Vertical phase tilt and damping effects

We next examine wave behavior in a parameter region where subcritical instability does not occur, but both O(1) and small-wave-amplitude solutions exist for the same parameter setting. By examining the difference between these two solutions, we aim to gain more insight into the O(1)-amplitude equilibration within the subcritical part of parameter space. For the parameter setting , two solutions exist, that is, multiple equilibria⁵ take place (Fig. 2a). Because κ_M = κ_T, term I equals zero, but for 0.26 < β ≤ 0.335, F₁ strongly impacts the zonal mean flow, since > . Moreover, the contribution of − to term II is much smaller than that of −U_2,yy (Fig. 2b), indicating that the abrupt transition from the O(1) to the small-amplitude wave solution cannot be attributed solely to the difference in the baroclinicity of the modified zonal mean flow. Instead, other effects, such as damping of the eddies and a reduction in wave activity absorption in the upper layer (as suggested by LH), also need to be taken into account. To see this, first consider the eddy potential enstrophy equation:

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where

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and j = 1, 2 once again denotes the upper and lower layers, respectively. Taking a time mean, (8) can be written as

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where _j = , which represents eddy enstrophy production by the transient zonal mean PV flux.

For a steady flow (T_j = 0), which is the case for all of the runs shown in Figs. 1 and 2, and for a single zonal wave (J_j = 0), the above equation reduces to

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Focusing on the lower layer, this relation indicates that if |∂Q₂/∂y| is small, as would be the case if F₂ is to have a finite amplitude in weakly supercritical and subcritical regions [as can be seen from (7)], the finite-amplitude wave can be sustained only if ₂ is sufficiently small. Ignoring the subgrid-scale diffusion term, ₂ can be rewritten as

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The first and third terms within both sets of parenthesis on the rhs of (10) contribute to positive ₂ (therefore these terms damp the eddy enstrophy). In contrast, for the second term in both parentheses, if the phase difference between ψ*₁ and ψ*₂ is small so that the contribution of these terms is large, the sign of ₂ can be negative, thus leading to an increases in the eddy enstrophy. Of course, if ψ*₁ and ψ*₂ are exactly in phase, the eddy heat flux becomes identically zero, and the flow cannot support the presence of eddies. Thus, the phase difference, δϕ, between ψ*₁ and ψ*₂ should be small for ₂ to be either weakly positive or even negative, but δϕ cannot be too small because otherwise waves could not be generated.

The above process is presented schematically in Fig. 3. The thermal contribution to the second term on the rhs of (10) (top line) can be written as κ_M. When δϕ is small, κ_M is negative (Figs. 3b and 3c), and the impact of the Ekman pumping is to increase |ψ̂*| and decrease |∇²ψ*₂|. If the change in |ψ̂*| dominates, κ_M will become even more negative, resulting in a lowering of D₂. In contrast, if δϕ is large, κ_M is positive (Figs. 3e and 3f), and the Ekman pumping will decrease both |ψ̂*| and |∇²ψ*₂|. which results in κ_M remaining positive.

Comparison between wave fields in the upper and lower branches of the solution for shows that δϕ is markedly smaller for the upper branch than for the lower branch (see Fig. 4; the upper and lower branches correspond to the O(1) and the small-amplitude solutions in Fig. 2a.) The quantity δϕ can be estimated visually from Fig. 4 by comparing the middle panel with the corresponding lower panel.

Returning to the κ_M and κ_T parameter settings of Fig. 1, we display in Fig. 5 solutions for (left) and (right) , representing the supercritical and subcritical regions of the parameter space, respectively. As can be seen, the supercritical and subcritical solutions, which both are of O(1) amplitude, have a small δϕ, as in the upper branch solution in Fig. 4. Figure 2c indicates that δϕ is much smaller for all finite-amplitude solutions than for the small amplitude solutions.

By comparing the potential enstrophy budgets in the and runs (see Figs. 1 and 2), we can gain insight into the manner by which frictional dissipation increases the amplitude of equilibrated wave. We begin by examining the lower-layer budget (see Fig. 6). For both parameter settings, as β increases, there is a rapid weakening in the Ekman damping. For , where subcritical instability does not occur, the Ekman damping term is always negative. In contrast, for , if β > 0.14, the Ekman damping term becomes positive, acting to generate, rather than to dissipate, the lower-layer eddy potential enstrophy! As a result, in spite of the rapid decrease in the generation term, −[υ*₂q*₂](∂Q₂/∂y), the eddy enstrophy remains at an almost constant value with respect to β until the subcritical instability ceases to exist.

We next examine the upper-layer potential enstrophy budget. For , the eddy enstrophy generation, −[υ*₁q*₁](∂Q₁/∂y), is almost constant with respect to β. This is because, as β increases, while |[υ*_jq*_j]| becomes weaker, (∂Q₁/∂y) becomes greater. This behavior does not occur for , where the subcritical instability does not take place.

Further insight into the role of the Ekman pumping and its relationship with enstrophy generation can be gained by examining the transient evolution of key terms in the budget. Figure 7 displays the initial transient evolution of selected fields for . This case is chosen because it is supercritical and also closest to the stability boundary. The initial state is comprised of the radiative equilibrium state and an infinitesimal amplitude arbitrary perturbation. By day 100, the most unstable normal mode emerges, and the normal mode is maintained until about day 250 (see Fig. 8b). This can be inferred from Figs. 7b and 7f, since the zonal mean ∂Q_j/∂y does not undergo noticeable changes until day 250. Afterward, the upper- and lower-layer PV fluxes (Figs. 7c and 7g) begin to change from their normal mode form. Thus, the contours in Figs. 7c and 7g represent the eddy PV fluxes after they have deviated from their normal mode form.

As the perturbation starts to modify the mean flow, the enstrophy generation in both layers becomes increasingly confined toward the jet center (Figs. 7a and 7e). More importantly, after day 370, while −[υ*₁q*₁](∂Q₁/∂y) remains strong, −[υ*₂q*₂](∂Q₂/∂y) substantially weakens. As a result, |q*₁| becomes disproportionally larger than |q*₂|. (Prior to day 250, |q*₁|/|q*₂| is approximately 1.7; by day 450, this ratio equals to 3.4.) This inequality allows δϕ to become small (the right columns in Fig. 8 show that δϕ drops between day 250 and day 450), since q*₁ plays the dominant role in inducing both ψ*₁ and ψ*₂ (Hoskins et al. 1985). By day 450, ψ*₂ is induced mostly by q*₁ (compare the upper-layer PV field in Fig. 8i with the lower-layer eddy streamfunction field in Fig. 8j). According to the foregoing discussion regarding Fig. 3, as δϕ decreases, the Ekman pumping contribution will become less negative, and in some cases can be positive. Indeed, Fig. 7h shows that the negative Ekman pumping contribution starts to weaken at around day 370, when −[υ*₂q*₂](∂Q₂/∂y) also starts to weaken. After day 390, the Ekman pumping contributes toward a growth of ϵ₂.

Because the foregoing discussion suggests that growth by Ekman pumping requires −[υ*₁q*₁](∂Q₁/∂y) to be sufficiently larger than −[υ*₂q*₂](∂Q₂/∂y) so that |q*₁| ≫ |q*₂|, we next examine how this inequality develops. Initially, before day 250, [υ*₁q*₁] (Figs. 7c) is dominated by the vorticity flux, [υ*₁ζ*₁] (Fig. 7d), whose minima occur at the critical latitudes. However, as the waves amplify and modify the mean flow, both the critical latitudes and the two [υ*₁ζ*₁] minima shift toward the jet center. As the two negative [υ*₁q*₁] centers continue their jetward march, between days 330 and 390, ∂²[υ*₁q*₁]/∂y² becomes increasingly negative in the jet core region (Fig. 7j). Concurrent with this increase, Û also intensifies (Fig. 7k). This relationship between ∂²[υ*₁q*₁]/∂y² and Û is consistent with the theory⁶ of Robinson (2006).

Conforming to the above interpretation, during the same time period rapid growth occurs in [υ*₂q*₂] (Fig. 7g), which stabilizes the flow. Since the heat flux, which is almost identical to [υ*₂q*₂], is maximized at the jet center, this increase in the heat flux acts to shift the two [υ*₁q*₁] minima even closer to the jet center. As a result, large −[υ*₁q*₁] can occur where ∂Q₁/∂y is also large (cf. Figs. 7b and 7c), allowing for a large production of |q*₁| in the jet core region. By contrast, the enstrophy generation in the lower layer is much smaller because ∂Q₂/∂y weakens substantially (Fig. 7f). This weakening in ∂Q₂/∂y is due to the growth of [υ*₂q*₂] described earlier. Of course, the upper-layer PV flux also weakens ∂Q₁/∂y, but the sum of the nonzero β and the jetlike U_e,1 keeps |∂Q₁/∂y| much greater than |∂Q₂/∂y|.

The occurrence of large |q*₁| can also be interpreted in terms of energetics. As described earlier, the wave initially breaks at the critical latitude. During this stage, since ∂Q₁/∂y > 0 at the critical latitude, the wave is absorbed and the barotropic energy conversion, defined as −[u*₁υ*₁](∂U₁/∂y), is active. However, as the jetward shift occurs in both the critical latitude and the wave breaking latitude (indicated by the negative bands in [υ*₁ζ*₁]; Fig. 7d), the original critical latitude ceases to exist. In spite of this cessation of the critical latitude, the wave breaking continues to occur at the same latitude as before (Fig. 7d). Comparing Figs. 7d and 7b, it can be seen that this off-critical latitude wave breaking occurs at latitudes where ∂Q₁/∂y has a local minimum. Because this local ∂Q₁/∂y minimum is associated with small values of ∂U₁/∂y (as can be inferred from Fig. 7i), the barotropic energy conversion in this region is muted. Indeed, the barotropic energy conversion, normalized by total eddy energy, is found to decrease by a factor of 4 between day 250 and day 450 (not shown). Similar behavior in barotropic energy conversion was observed by Lachmy and Harnik (2009), who attributed their O(1) equilibration to the weakened barotropic energy conversion.

In LH, the subcritical instability was explained in terms of wave reflection at the critical latitude, since in the case that they examined ∂Q₁/∂y = 0 at the critical latitude. However, in the multiwave model, even for the subcritical case, ∂Q₁/∂y ≠ 0 at the critical latitude (not shown). The above finding suggests that an important ingredient for finite-amplitude equilibration is a muted barotropic decay, caused by critical latitude wave breaking with a zero or a small ∂Q₁/∂y. This discussion will be revisited in section 3c.

To test if the location of the maximum [υ*₁q*₁] is indeed the key factor for the wave amplification in the subcritical region of the parameter space, three calculations are performed with . As summarized by Fig. 1, at this parameter setting the final equilibrium wave amplitude is zero, even if the initial condition is a steady wave solution with a smaller value of β. As can be seen in Figs. 9a and 9b, when initialized by the equilibrium steady wave solution for , the wave decays. In the second and third calculations, the initial condition is once again the equilibrium steady wave solution for , but Q₁ is integrated with a prescribed [υ*₁q*₁]. The remaining three model variables, Q₂, q*₁, and q*₂, are integrated forward in time in the standard manner. This prescribed [υ*₁q*₁] field for the first calculation, which corresponds to that for β = 0.37, is shown in Fig. 10. For brevity, this calculation will be referred to as EXP1. It can be seen from Figs. 9c and 9d that when the two minima in [υ*₁q*₁] remain on the jetward side of the critical latitudes (indicated by thick contours), a finite-amplitude wave is maintained. For the second calculation (hereafter EXP2), the prescribed [υ*₁q*₁] profile was altered so that the [υ*₁q*₁] minima are shifted away from the jet center (dotted curve in Fig. 10). In this calculation, the wave field rapidly decays to zero (Figs. 9e and 9f).

The time evolution of the potential enstrophy budget, (8), for EXP1 (Figs. 11a,b), supports the earlier interpretation that the upper-layer enstrophy generation leads the amplification by Ekman pumping. Figure 11a shows the budget terms normalized by ϵ_j. It can be seen that the rise in ϵ₁ by the upper-layer generation term is followed 15 days later by an enhancement in ϵ₂ by the Ekman pumping. Furthermore, this growth by the Ekman pumping is followed by an increase in ϵ₂, indicating that the Ekman pumping contribution is not a response to a change in ϵ₂, say, by the generation term. This interpretation is also borne out in EXP2 (Figs. 11c,d). It can be seen that while both the upper-layer generation term and the Ekman pumping contribution rapidly decrease, accounting for the decay of ϵ₁ and ϵ₂, respectively, the growth by the lower-layer generation term remains positive and even strengthens (Figs. 11c,d). Once again this behavior indicates that the Ekman pumping takes on a lead role in the lower-layer eddy enstrophy evolution.

c. The relationship between wave breaking latitude and critical latitude

The previous subsection indicates that the proximity of the wave breaking latitude to the jet center is the key for the finite-amplitude equilibration of the wave field. The transient wave evolution (Fig. 7) shows that wave breaking starts at the critical latitude, but as the waves grow, jetward shifts occur in both the critical latitudes and the wave breaking latitudes. As indicated by Fig. 7d and the PV fields in Fig. 8, this jetward shift is accompanied by a widening of the wave breaking region. In addition, because the wave breaking persists even after the original critical latitudes terminate (approximately at day 430), the central latitude of the wave breaking region ends up being located notably closer to the jet center than the newly formed (at about day 380) critical latitudes.

This transient evolution sheds light onto the relationship between the wave breaking latitude and the critical latitude. It can be seen from Fig. 12a that for β ≤ 0.29, the wave breaking latitude, defined by the latitude of the local minimum in U₁ (denoted by y_m hereafter), is located markedly closer to the jet center than either y_c or the critical latitude of the radiative equilibrium flow (denoted by y_c,e). The transient evolution described in the previous paragraph indicates that this difference between the wave breaking and critical latitudes arises because the wave breaking persists even after the original critical latitudes desist. This persistence could be due to an increase in |∂U₁/∂y| (Fig. 7i), since it has been shown that in the presence of meridional shear, finite-amplitude waves can break before they reach their critical latitudes (Held and Phillips 1987; Robinson 1988; Feldstein and Held 1989; Fyfe and Held 1990; Gong and Feldstein 2009, submitted to J. Atmos. Sci.).

Figure 12a also indicates that farther into the subcritical part of the parameter space, the wave breaking is still closer to the jet center than is y_c,e. However, it can be seen that y_c has shifted much closer to the jet center, resulting in the overlap of the wave breaking latitude and y_c. This implies that for these values of β the critical latitudes form as a result of the wave breaking (cf. Fyfe and Held 1990). In these cases where y_m ≈ y_c, we see that Q_1,y is close to zero (Fig. 12b). According to inviscid critical layer theory (e.g., Killworth and Mcintyre 1985), when Q_y = 0 the wave is reflected rather than absorbed. Consistent with this theory, meridional tilt of the eddies is absent at the critical layer (see Fig. 5e) and barotropic decay is still much reduced. This suggests that for subcritical β, the weakening of the barotropic decay resulting from zero Q_1,y at y_c may also contribute to the O(1) finite-amplitude equilibration (LH).