## 1. Introduction

Addressing the question of how baroclinic waves equilibrate is pivotal for understanding the strength of both the time-mean midlatitude storm tracks and the meridional temperature gradient. There have been several different approaches for treating this problem: weakly nonlinear theory (Pedlosky 1970), homogeneous turbulence (Charney 1971; Salmon 1980; Herring 1980; Held and Larichev 1996), nonlinear Liapunov stability theory (Shepherd 1988), and laboratory experiments (Fultz et al. 1959).

The appeal of the first three approaches is their ability to express answers in an analytical form that provides a clean framework for better understanding the complex nonlinear problem. However, the benefit to these approaches is limited by the use of assumptions that are necessary for obtaining analytical solutions. For example, our experience with daily weather indicates that nonlinear wave–wave interactions are strong, an observation that is inconsistent with the central assumption of weakly nonlinear theory. However, the assumption of homogeneous turbulence is at odds with the prominence of westerly jets and attendant nonzero time-mean turbulent eddy vorticity fluxes (Robinson 2006); to use the nonlinear stability theorem, Shepherd (1988) assumed that the flow in question is subject to potential vorticity damping, which implies that the top of the atmosphere is subject to the same Ekman damping rate as the bottom of the atmosphere. Although these papers have presented much insight into general equilibration processes for baroclinic waves, because the model settings are not realistic for the atmosphere, it remains an open question as to what processes determine the *O*(1) amplitude of waves in the atmosphere.

Many of the above studies used highly idealized versions of the two-layer quasigeostrophic model (e.g., Pedlosky 1970; Salmon 1980; Held and Larichev 1996; Shepherd 1988). However, if more realistic features are included, such as relaxation of the zonal mean zonal wind toward a jetlike wind profile and dissipation by lower-layer Ekman pumping, the two-layer model can produce many of the observed large-scale midlatitude flow characteristics, including teleconnection-like stationary waves (Speranza and Malguzzi 1988), barotropic decay (Feldstein and Held 1989), and baroclinic wave packets (Lee and Held 1993), as well as the zonal index (Lee and Feldstein 1996b). Furthermore, for such a model setting, subcritical^{1} instability occurs within a parameter space spanned by nondimensional^{2} *β* and the Ekman damping coefficient, *κ _{M}* (Lee and Held 1991, hereafter LH). For example, if the value of

*κ*is fixed and the value of

_{M}*β*is gradually increased, the model’s baroclinic waves tend to equilibrate at an

*O*(1) amplitude in the subcritical region of the model’s parameter space. The only exception to this behavior occurs in the parameter region where

*κ*is very small (see Fig. 3 of LH). LH offered a physical explanation for this behavior in terms of wave reflection at critical latitudes, where the wave’s phase speed equals the zonal mean flow speed. In this picture, the waves are first absorbed at the critical latitudes. This wave absorption then reduces the basic-state potential vorticity gradient to a value of zero at the critical latitudes, thereby turning the absorbing critical latitudes into reflecting surfaces where wave activity can no longer be dissipated.

_{M}The behavior of the model’s baroclinic waves and the zonal mean flow may be more realistic within the subcritical and weakly supercritical parts of its parameter space than in the strongly supercritical part. This is because the observed time-mean state of the atmosphere is close to being marginally stable (Hall and Sardeshmukh 1998), yet the amplitude of the observed baroclinic waves is of *O*(1). Moreover, as will be shown later, a common feature in the model’s subcritical region is what is known as a *self-maintaining jet*^{3} (Robinson 2006). He showed that at the jet center, the eddy-driven mean meridional circulation strengthens the vertical wind shear to a value beyond that associated with radiative equilibrium. This was shown to occur through the eddy momentum flux, not the eddy heat flux. As we will see, because the self-maintaining jet is most often found in the marginally unstable and subcritical region of the model’s parameter space rather than in its more supercritical state, we are further motivated to examine the wave equilibration in this part of the parameter space.

The goal of this study is to investigate the physical processes that drive the equilibration of *O*(1)-amplitude baroclinic waves in a jetlike flow, focusing on the weakly supercritical and subcritical regions of the model’s parameter space. We concentrate on this part of the parameter space because, as discussed above, it is these parameter settings that yield results that qualitatively most closely resemble the atmosphere.

In this study, we will explore an alternative mechanism of finite-amplitude wave equilibration that involves the impact of Rossby waves that break before they reach their critical latitude (where the wave phase speed equals the background zonal mean zonal wind speed). As we will see, when wave breaking exhibits this property, the vertical phase tilt is reduced. This results in 1) an increase in the upper-layer wave generation and 2) a weakening of the lower-layer wave damping. Together, these processes can account for the *O*(1)-amplitude wave equilibration.

A brief description of the model, together with an analysis of the model equations, is presented in section 2. Results from selected numerical model calculations are shown in section 3, and the conclusions follow in section 4.

## 2. Model equation analysis

*β*plane, with flat rigid boundaries on the top and the bottom. This model is identical to that used by LH, wherein the dimensionless governing potential vorticity equations arewhere the potential vorticity

*q*satisfiesThe subscripts

_{j}*j*= 1 and 2 refer to the upper and lower layers, respectively,

*ψ*is streamfunction, and

_{j}*ψ̂*≡ (

*ψ*

_{1}−

*ψ*

_{2})/2. The model is driven toward a prescribed thickness field

*ψ̂*≡ (

_{e}*ψ*

_{e,1}−

*ψ*

_{e,2})/2, which is analogous to the radiative equilibrium temperature field. For simplicity, it is assumed that

*u*

_{e,2}≡ −∂

*ψ*

_{e,2}/∂

*y*= 0 everywhere. The coefficients

*κ*and

_{T}*κ*are the thickness and Ekman damping rates, respectively. To emphasize its analogous role in the atmosphere, the thickness damping will be referred to hereafter as thermal damping. The term

_{M}*ν*∇

^{6}

*ψ*represents the enstrophy cascade toward subgrid scales.

_{j}Regardless of the role played by the critical latitudes (LH) or by the eddy-driven mean meridional circulation^{4} (Robinson 2006), for waves of nonzero amplitude to be maintained against dissipation, there must be a wave activity source. The time-mean wave activity source is proportional to the time-mean potential vorticity flux [e.g., (7.23) in Vallis 2006]. Since wave activity is generated in the lower layer for baroclinically unstable flows, we examine the lower-layer potential vorticity flux.

*U*is the zonal mean zonal wind and

_{j}*Q*is the zonal mean potential vorticity. Taking the derivative of (2.1) with respect to

_{j}*y*yieldsThe expression (3) embodies the idea of the self-maintaining jet of Robinson (2006). Ignoring the diffusion term, (3) states that if ∂

^{2}

*y*

^{2}< 0, the time-mean zonal wind shear is greater than the radiative equilibrium shear. At the center of the jet, where the eddy heat flux is typically poleward, (3) indicates that a self-maintaining jet is associated with a strong eddy momentum flux convergence in the upper layer.

*U*

_{2}/∂

*y*by adding (2.1) and (2.2), followed by a rearrangement of terms:Because the scale selective dissipation terms are included in the numerical model to deal with the enstrophy cascade toward subgrid scales, ideally these terms should be negligible. As we will see in the next section, for the parameter space of interest, the contribution by these diffusion terms is small.

*Q*

_{2}/∂

*y*rather than

_{2}and

_{1}, if we assume that the meridional structure of

_{1}and

_{2}take the form

_{1,yy}/

_{1}= −

*l*

_{1}

^{2}(

*y*) and

_{2,yy}/

_{2}= −

*l*

_{2}

^{2}(

*y*), after minor rearrangement, (5) can be rewritten aswhere

*ξ*is the supercriticality of the radiative equilibrium state,andAlternatively, the rhs of (6) can be expressed asWe next discuss the terms on the rhs of (7). As will be shown in the next section, the diffusion term in (7) is very small. If

_{e}*l*

_{1}

^{2}

_{1}> 0 at the jet center, as is necessary for the self-maintaining jet (Robinson 2006), and as is the case for the Northern Hemisphere midlatitude winter flow [implied by Fig. 6 of Edmon et al. (1980)], then as long as

*κ*>

_{M}*κ*, which is typical for the atmosphere,

_{T}_{1,yy}is proportional to the lower-layer PV flux. Because the foregoing analysis is based on a steady state, causality between

_{1}and

_{2}cannot be established. Nevertheless, the proportionality between

*l*

_{1}

^{2}

_{1}and

_{2}may be interpreted as follows: if

*l*

_{1}

^{2}

_{1}acts to destabilize the mean flow [by (3)], then to maintain a constant supercriticality ∂

*Q*

_{2}/∂

*y*,

_{2}must respond to stabilize the flow. As will be described in section 3b, an example of transient wave evolution (Fig. 7) supports this interpretation.

The condition *κ _{M}* >

*κ*, together with the condition

_{T}*l*

_{1}

^{2}

_{1}> 0, can be understood as representing competition between two terms in (5), the destabilizing effect by the vertical shear,

*κ*=

_{T}*κ*, these two effects are precisely canceled, and

_{M}_{1}has no impact on either

_{2}or the supercriticality, −∂

*Q*

_{2}/∂

*y*.

The negative _{1,yy} arises from eddy momentum flux convergence at the jet center and divergence at the jet flanks. Noting that the magnitude of _{1,yy} is proportional to *l*_{1}^{2}, which is related to the width of the jet, because large-eddy momentum flux divergence typically coincides with wave breaking, this relation predicts that—all else being equal—larger _{2} would take place if wave breaking occurs closer to the jet center in the upper layer. This will be revisited later in section 3.

_{1,yy}and the supercriticality can be examined by recalling (5), which indicates that in order for the wave-modified zonal mean state to be more supercritical than the radiative equilibrium state (see the discussion in the introduction)—that is, for ∂

*Q*

_{2}/∂

*y*< ∂

*Q*

_{e,2}/∂

*y*=

*β*−

*U*

_{e,2}= 0)—thenTherefore, even when [∂

^{2}

*y*

^{2}] < 0 (cf. Robinson 2006), in order for (∂

*Q*

_{2}/∂

*y*) < (∂

*Q*

_{e,2}/∂

*y*), the Ekman damping must be stronger than the thermal damping [1 − (

*κ*/

_{M}*κ*) < 0] and |[∂

_{T}^{2}

*y*

^{2}]| < |[1 − (

*κ*/

_{M}*κ*)] [∂

_{T}^{2}

*y*

^{2}]|, since ∂

^{2}

*y*

^{2}is usually negative. As will be discussed in the next section, at least within the parameter space that we examine, the above inequality is never satisfied.

## 3. Numerical model calculations

_{2}. To exploit the results of LH (see Fig. 3 in that paper), the choice for the model parameters is made to be identical with those of that study. The only exceptions are the meridional resolution of the model and the value of

*ν*. The width of the channel is set equal to 30, there are 200 grid points across the channel, and the value of

*ν*is 5 × 10

^{−4}. For completeness, the remaining parameters are briefly described below. The structure of

*τ*is chosen so that the corresponding vertical shear of the zonal wind takes on a Gaussian form:where

_{e}*y*= 0 is at the middle of the channel, and

*σ*

^{2}= 10. Unless stated otherwise, the value of

*κ*is fixed at

_{T}It was found by LH that with only a single zonal wave (if it is chosen to be the most unstable wave for the radiative equilibrium profile) this model was capable of capturing the qualitative behavior, including the subcritical instability, of the statistically steady state of the corresponding full spectral model. Given this property, because the diagnostics to be presented below are much simpler, we use a single wave model as the primary tool for the numerical calculations. For the parameter space of interest, the most unstable zonal wavenumber is *k* = 0.8, so all of our single wave calculations are performed with this zonal wavenumber. A few selected calculations also make use of a full spectral model composed of 20 zonal waves, ranging from *k* = 0.1 to 2.0.

### a. The relationship between F _{1,yy} and supercriticality

As stated earlier, the relative contribution of the terms on the rhs of (7) toward _{2} 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,

*S*

_{1}, in Fig. 1a, which displays

_{2}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 *S*_{1}, Fig. 1a displays the sum of the rhs of (7) (denoted by *S*_{2}), and the sum of the rhs of (7) excluding the diffusion term (denoted by *S*_{3}). Because the semi-implicit scheme is used to calculate the damping terms, the terms *S*_{1} and *S*_{2} are very similar, but not identical. Figure 1a shows that *S*_{1} and *S*_{2} are essentially identical. More importantly, because the contribution by the _{z} term is less than 5% (cf. *S*_{3} with *S*_{2}), 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 _{2} 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 _{1,yy} = −*l*_{1}^{2}_{1}), *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, [(∂

*Q*

_{2}/∂

*y*) − (∂

*Q*

_{e2}/∂

*y*)] (hereafter called term II) is always positive. This is because the effect of term I, which is proportional to

*Û*, is offset by the influence of the horizontal curvature term, −

_{e}*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, (∂

*Q*

_{2}/∂

*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

*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

^{5}take place (Fig. 2a). Because

*κ*=

_{M}*κ*, term I equals zero, but for 0.26 <

_{T}*β*≤ 0.335,

_{1}strongly impacts the zonal mean flow, since

*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:whereand

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

_{j}=

_{j}= 0), which is the case for all of the runs shown in Figs. 1 and 2, and for a single zonal wave (

_{j}= 0), the above equation reduces toFocusing on the lower layer, this relation indicates that if |∂

*Q*

_{2}/∂

*y*| is small, as would be the case if

_{2}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 asThe 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,

*δ*

*ϕ*, between

*ψ**

_{1}and

*ψ**

_{2}should be small for

_{2}to be either weakly positive or even negative, but

*δ*

*ϕ*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}*

*δ*

*ϕ*is small,

*κ*

_{M}*ψ̂**| and decrease |∇

^{2}

*ψ**

_{2}|. If the change in |

*ψ̂**| dominates,

*κ*

_{M}_{2}. In contrast, if

*δ*

*ϕ*is large,

*κ*

_{M}*ψ̂**| and |∇

^{2}

*ψ**

_{2}|. which results in

*κ*

_{M}Comparison between wave fields in the upper and lower branches of the solution for *δ**ϕ* 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 *δ**ϕ* can be estimated visually from Fig. 4 by comparing the middle panel with the corresponding lower panel.

Returning to the *κ _{M}* and

*κ*parameter settings of Fig. 1, we display in Fig. 5 solutions for (left)

_{T}*O*(1) amplitude, have a small

*δ*

*ϕ*, as in the upper branch solution in Fig. 4. Figure 2c indicates that

*δ*

*ϕ*is much smaller for all finite-amplitude solutions than for the small amplitude solutions.

By comparing the potential enstrophy budgets in the *β* increases, there is a rapid weakening in the Ekman damping. For *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**_{2}*q**_{2}](∂*Q*_{2}/∂*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 *υ**_{1}*q**_{1}](∂*Q*_{1}/∂*y*), is almost constant with respect to *β*. This is because, as *β* increases, while |[*υ*** _{j}q**

*]| becomes weaker, (∂*

_{j}*Q*

_{1}/∂

*y*) becomes greater. This behavior does not occur for

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 *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**_{1}*q**_{1}](∂*Q*_{1}/∂*y*) remains strong, −[*υ**_{2}*q**_{2}](∂*Q*_{2}/∂*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 *δ**ϕ* to become small (the right columns in Fig. 8 show that *δ**ϕ* 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 *δ**ϕ* 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**_{2}*q**_{2}](∂*Q*_{2}/∂*y*) also starts to weaken. After day 390, the Ekman pumping contributes toward a growth of *ϵ*_{2}.

Because the foregoing discussion suggests that growth by Ekman pumping requires −[*υ**_{1}*q**_{1}](∂*Q*_{1}/∂*y*) to be sufficiently larger than −[*υ**_{2}*q**_{2}](∂*Q*_{2}/∂*y*) so that |*q**_{1}| ≫ |*q**_{2}|, we next examine how this inequality develops. Initially, before day 250, [*υ**_{1}*q**_{1}] (Figs. 7c) is dominated by the vorticity flux, [*υ**_{1}*ζ**_{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**_{1}*ζ**_{1}] minima shift toward the jet center. As the two negative [*υ**_{1}*q**_{1}] centers continue their jetward march, between days 330 and 390, ∂^{2}[*υ**_{1}*q**_{1}]/∂*y*^{2} becomes increasingly negative in the jet core region (Fig. 7j). Concurrent with this increase, *Û* also intensifies (Fig. 7k). This relationship between ∂^{2}[*υ**_{1}*q**_{1}]/∂*y*^{2} and *Û* is consistent with the theory^{6} of Robinson (2006).

Conforming to the above interpretation, during the same time period rapid growth occurs in [*υ**_{2}*q**_{2}] (Fig. 7g), which stabilizes the flow. Since the heat flux, which is almost identical to [*υ**_{2}*q**_{2}], is maximized at the jet center, this increase in the heat flux acts to shift the two [*υ**_{1}*q**_{1}] minima even closer to the jet center. As a result, large −[*υ**_{1}*q**_{1}] can occur where ∂*Q*_{1}/∂*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 ∂*Q*_{2}/∂*y* weakens substantially (Fig. 7f). This weakening in ∂*Q*_{2}/∂*y* is due to the growth of [*υ**_{2}*q**_{2}] described earlier. Of course, the upper-layer PV flux also weakens ∂*Q*_{1}/∂*y*, but the sum of the nonzero *β* and the jetlike *U*_{e,1} keeps |∂*Q*_{1}/∂*y*| much greater than |∂*Q*_{2}/∂*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 ∂*Q*_{1}/∂*y* > 0 at the critical latitude, the wave is absorbed and the barotropic energy conversion, defined as −[*u**_{1}*υ**_{1}](∂*U*_{1}/∂*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**_{1}*ζ**_{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 ∂*Q*_{1}/∂*y* has a local minimum. Because this local ∂*Q*_{1}/∂*y* minimum is associated with small values of ∂*U*_{1}/∂*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 ∂*Q*_{1}/∂*y* = 0 at the critical latitude. However, in the multiwave model, even for the subcritical case, ∂*Q*_{1}/∂*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 ∂*Q*_{1}/∂*y*. This discussion will be revisited in section 3c.

To test if the location of the maximum [*υ**_{1}*q**_{1}] is indeed the key factor for the wave amplification in the subcritical region of the parameter space, three calculations are performed with *β*. As can be seen in Figs. 9a and 9b, when initialized by the equilibrium steady wave solution for *Q*_{1} is integrated with a prescribed [*υ**_{1}*q**_{1}]. The remaining three model variables, *Q*_{2}, *q**_{1}, and *q**_{2}, are integrated forward in time in the standard manner. This prescribed [*υ**_{1}*q**_{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**_{1}*q**_{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**_{1}*q**_{1}] profile was altered so that the [*υ**_{1}*q**_{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

*ϵ*

_{1}by the upper-layer generation term is followed 15 days later by an enhancement in

*ϵ*

_{2}by the Ekman pumping. Furthermore, this growth by the Ekman pumping is

*followed*by an increase in

*ϵ*

_{2}, indicating that the Ekman pumping contribution is not a response to a change in

*ϵ*

_{2}, 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

*ϵ*

_{1}and

*ϵ*

_{2}, 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 *U*_{1} (denoted by *y _{m}* hereafter), is located markedly closer to the jet center than either

*y*or the critical latitude of the radiative equilibrium flow (denoted by

_{c}*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 |∂

*U*

_{1}/∂

*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*. This implies that for these values of

_{c}*β*the critical latitudes form as a result of the wave breaking (cf. Fyfe and Held 1990). In these cases where

*y*≈

_{m}*y*, we see that

_{c}*Q*

_{1,y}is close to zero (Fig. 12b). According to inviscid critical layer theory (e.g., Killworth and Mcintyre 1985), when

*Q*= 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

_{y}*β*, the weakening of the barotropic decay resulting from zero

*Q*

_{1,y}at

*y*may also contribute to the

_{c}*O*(1) finite-amplitude equilibration (LH).

## 4. Conclusions

This study used a two-layer QG model to study mechanisms that drive the finite-amplitude equilibration of baroclinic waves on a jet. It was found that the finite-amplitude equilibration hinges on wave breaking occurring sufficiently close to the jet center. In the supercritical and in part of the subcritical regions of the parameter space, the wave breaking takes place before the waves reach their critical latitudes. This “jetward” wave breaking results in the occurrence of finite-amplitude wave equilibration through the following processes. First, from a potential vorticity perspective, the reduction in the meridional scale of the eddies, arising from the jetward breaking, results in −∂^{2}[*υ**_{1}*q**_{1}]/∂*y*^{2} becoming increasingly negative in the jet core region. This process, which destabilizes the mean flow,^{7} is followed by an enhanced lower-layer PV flux that reduces |∂*Q*_{2}/∂*y*| and thereby limits the growth of |*q**_{2}|. The same jetward wave breaking also concentrates the upper-layer PV flux toward the jet center where ∂*Q*_{1}/∂*y* is largest. This results in a disproportionally strong intensification of |*q**_{1}| relative to |*q**_{2}|. Because |*q**_{1}| is much greater than |*q**_{2}|, *q**_{1} plays the dominant role in inducing both *ψ**_{1} and *ψ**_{2}, and as a result the vertical phase tilt, *δ**ϕ*, is small. This results in a weakening of the Ekman damping, and for some parameter settings the Ekman pumping can even act as a wave source, contributing toward *O*(1)-amplitude wave equilibration.

From the equilibration processes described above, we arrive at the interesting conclusion that the amplification by Ekman pumping hinges on |*q**_{1}| ≫ |*q**_{2}|, and that this inequality requires a large |∂*Q*_{1}/∂*y*| and a small |∂*Q*_{2}/∂*y*|. This implies that a state of weak supercriticality (which led to the idea of baroclinic adjustment; Stone 1978) can be regarded as both a symptom and a necessary ingredient of this dissipative amplification.

In the subcritical region of the parameter space, with even larger values of *β*, the wave breaking latitude and the critical latitude coincide. However, our analysis indicates that this particular critical latitude forms due to the wave breaking, rather than the other way around. The critical latitude that forms through this process has ∂*Q*_{1}/∂*y* ≈ 0. Thus, whether or not the wave breaking occurs at the critical latitude, all of the finite-equilibrium solutions found in this study are characterized by reduced barotropic decay.

According to the above argument, if wave breaking were to occur farther away from the jet center—say, at an absorbing critical latitude—then *δ**ϕ* would be relatively large (cf. Figs. 3d,e with Figs. 3a,b). This results in a strengthening of the Ekman pumping and thus a reduction of the lower-layer eddy potential enstrophy (Fig. 3f). At the same time, since the eastward acceleration of the zonal mean flow would occur over a broader latitudinal range, the resulting increase in ∂*Q*_{1}/∂*y* at the jet center would also be smaller than that in the situation depicted in Fig. 3a. Both of these processes hinder the maintenance of finite-amplitude waves. In fact, at least in the model used in this study, the coincidence of critical-latitude wave breaking, relatively large *δ**ϕ*, and small ∂*Q*_{1}/∂*y* results in either small-amplitude wave equilibration or a decay toward a zero-wave amplitude.

An additional set of experiments supports the importance of the jetlike equilibrium wind profile and the jetward wave breaking for the finite-amplitude wave equilibration. Although not shown in this paper, if the same two-layer QG model is driven toward a state where *U*_{e,1} takes on a top hat–like profile (Lee 1997), rather than toward an externally imposed jet, as in this study, subcritical instability does not occur. In those experiments, the final equilibrium states consist of multiple jets, but these jets are spontaneously generated by eddy fluxes, and the latitude of the wave breaking coincides with the critical latitudes (see Fig. 5b in Lee 1997).

In the atmosphere, since the zonal mean flow on the equatorward side of the jet is driven (externally^{8}) by an overturning Hadley circulation (Palmén and Newton 1969; Held and Hou 1980), one implication of this study is that the observed midlatitude storm track amplitude may depend on the tropical–subtropical upper tropospheric zonal mean zonal wind and PV gradient.

Observational evidence indicates that the state of the atmosphere is closer to the situation depicted in Fig. 3a than to the one sketched in Fig. 3d, suggesting that the finite-amplitude wave equilibration mechanism presented in this study may be relevant. In the Southern Hemisphere, as shown in Fig. 2 of Kim and Lee (2004), the central latitude of the wave breaking in the subtropics is typically 5°–10° poleward of the critical latitude,^{9} similar to the jetward wave breaking observed in our two-layer model. Because the results of this study are based mostly on a single-wave model, it is necessary that the model results be tested with a full wave spectrum. In the atmosphere, and in numerical models with a full zonal wave spectrum, Rossby waves are, more often than not, organized into wave packets (Lee and Held 1993; Chang 1993; Berbery and Vera 1996). In a general circulation model, within the wave packet, the meridional wave activity flux was found to be greatest at the leading edge of the wave packet (see Fig. 11 of Lee and Feldstein 1996a). Therefore, it is anticipated that the process depicted in Figs. 3a–c will occur at the leading edge of the wave packet and thus may not be apparent if the flow is zonally averaged. To address this issue, in a future study we plan to examine finite-amplitude equilibration in baroclinic wave packets.

## Acknowledgments

This study was supported by the National Science Foundation under Grant ATM-0647776. The author acknowledges many valuable conversations and comments from Steven Feldstein and helpful comments from anonymous reviewers. The insightful comments of one of the reviewers are greatly appreciated. The author also acknowledges beneficial correspondence with Orli Lachmy and Nili Harnik.

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^{1}

A state is referred to being subcritical if none of its normal modes have a positive growth rate. It needs to be noted that in the subcritical state in question, the lower-layer PV gradient is still negative at the center of the jet, but the presence of Ekman and thermal damping keep the normal modes’ growth rate from being positive.

^{2}

The nondimensional *β* is defined as *β* = *β̃λ _{d}*

^{2}/

*U*, where

*β̃*is the dimensional

*β*,

*λ*the Rossby radius of deformation, and

_{d}*U*the velocity scale. As such, the nondimensional

*β*is a measure of baroclinicity.

^{3}

As pointed out by Robinson (2006), the characteristics of the observed transformed Eulerian mean meridional (Andrews and McIntyre 1976; Edmon et al. 1980) circulation suggest that a similar process may be taking place in the atmosphere.

^{4}

Throughout this paper, the terms “eddy” and “wave” are used interchangeably.

^{5}

Among the runs that we examined, only this combination of parameters exhibits multiple equilibria. As indicated by LH, the area of the parameter region where multiple equilibria take place expands as the meridional resolution increases and the value of *ν* decreases. Because we use the multiple equilibria only as a tool to investigate the equilibration process, and also because LH examined multiple equilibria in detail, we did not make an additional attempt to find more points in the parameter space where multiple equilibria occur.

^{6}

As was discussed earlier in this paper, his theory is based on an analysis of statistically steady states. Nevertheless, this theory appears to hold for the transient evolution shown in Fig. 7.

^{7}

Although the theory of Robinson (2006) is based on statistically steady state, transient evolution supports this interpretation.

^{8}

In both observations (Pfeffer 1981) and models (Becker et al. 1997; Kim and Lee 2001; Walker and Schneider 2006), eddies are shown to play an important role for driving the Hadley circulation. The eddy driving of the Hadley circulation occurs through heat fluxes, momentum fluxes, indirect eddy driving of diabatic heating, and the eddy influence on static stability (Kim and Lee 2001). While the eddy momentum flux intensifies the Hadley circulation, it does so by *decelerating* the tropical–subtropical upper tropospheric zonal wind. Therefore, although the eddies can strengthen the Hadley circulation, the tropical/subtropical zonal winds are not driven by the eddies. These simultaneous changes in the zonal wind and the Hadley circulation can be seen by comparing Figs. 1 and 2 of Kim and Lee (2001).

^{9}

Eddy momentum and Eliassen–Palm fluxes were shown by Randel and Held (1991), but it is difficult to precisely evaluate the eddy momentum flux convergence from these plots.