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  • Whitaker, J. S., and A. Barcilon, 1995: Low-frequency variability and wavenumber selection in models with zonally symmetric forcing. J. Atmos. Sci.,52, 491–503.

  • View in gallery

    Meridional profiles of the zonal-mean states and their stability characteristics associated with an f-plane life cycle experiment. (a)–(c) initial condition; (d)–(f) final state for k = 1.3; (g)–(i) final state for k = 1.0; (j)–(l) final state for k = 0.8. The zonal wavenumber k is normalized by the inverse Rossby radius L−1R. (left) The zonal-mean flow as a function of y (in LR); the lower-layer profile is dashed. Unit is 0.2Γ1 L2R (=10 m s−1). (middle) Same as the left column but for the zonal-mean PV gradients. Unit is Γ1 (=5 × 10−11 s−1 m−1). (right) Growth rates of the unstable modes for the mean states shown in the two left columns as a function of k. Unit is 0.2 Γ1 LR (=10−6 s−1). The arrows indicate the wavenumber corresponding to the length of the channel. S1, first meridionally symmetric mode (slightly tilted due to the flow asymmetry); A1, first antisymmetric mode; S2, second symmetric mode; A, stationary mode whose amplitude maximizes at both jet axes; B, two counterpropagating modes whose amplitude maximizes at either one of the jet axes; C, two counterpropagating modes whose structure is nearly antisymmetric about each jet axis.

  • View in gallery

    Same as Fig. 1 except for the meridionally symmetric initial condition. S1, first symmetric mode; S2, second symmetric mode; A1, first antisymmetric mode. (i) and (l) Modes with similar growth rates correspond to various combinations of meridional symmetries [i.e., symmetry and antisymmetry about the three axes of the jet in (g) and (j)].

  • View in gallery

    Same as Fig. 2 but for a vertically antisymmetric initial condition. The zonal-mean states in the bottom row are the time average between t = 210 and 240 (t is normalized by Γ−11L−1R = 2 × 104 s), and the growth rates in (f) pertain to this averaged state. S1, first symmetric mode; S2, second symmetric mode; A1, first antisymmetric mode.

  • View in gallery

    Same as Fig. 2 but for the beta-plane experiment. (top) Initial condition; (row 2) k = 1.3; (row 3) k = 1.0; (row 4) k = 0.75; (bottom) k = 0.63. (f), (i), (l), (o) Arrows labeled “1” indicate the zonal wavenumber corresponding to the channel length, whereas those labeled “2” indicate the wavenumber of the second zonal harmonic. S1, S2, and S3 are meridionally symmetric modes and A1 is the antisymmetric mode.

  • View in gallery

    Same as Figs. 4j and 4k but for half the value of beta.

  • View in gallery

    (a) A diagram showing the meridional arrangement of three PV discontinuities in two layers. (b) Zonal-mean flow that sustains the PV geometry shown in (a). Here L = 1.2LR is assumed. Solid curve, upper-layer flow; dashed curve, lower-layer flow; thin solid curve, average of the upper- and lower-layer flows. (c) Growth rates of the unstable mode for the above flow as a function of the zonal wavenumber and meridional separation. (d) Same as (a) but for a barotropically unstable situation in a single layer. (e) Zonal-mean flow that sustains the PV geometry shown in (d). (f) Same as (c) but for the flow defined by (d) and (e). (g) Same as (a) but with a beta effect. See text for details. (h) Zonal-mean flow for (g). (i) Same as (c) but for the flow defined by (g) and (h).

  • View in gallery

    A schematic of the PV mixing in the lower layer (beta plane). The thick solid lines indicate PV as a function of y. (a) The negative PV gradients (a jump) are meridionally isolated initially. (b) As PV is partially mixed, the negative PV jump at the edges diminishes in magnitude. (c) When the critical mixing length is reached, the negative PV jumps disappear and the flow becomes baroclinically neutral. Notice that in (5.4), the outer regions with the constant PV gradient are replaced by the ones with constant PV (dashed lines).

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Baroclinic–Barotropic Adjustments in a Meridionally Wide Domain

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  • 1 Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois
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Abstract

Baroclinic adjustment hypothesis fails to account for the enhancement of the barotropic jet observed in idealized baroclinic-wave life-cycle simulations. In this paper, an adjustment theory more consistent with the numerical results is developed through a careful examination of life-cycle experiments and nonseparable eigenvalue problems using the two-layer model.

In all cases examined, nonlinear eddies emanate from an unstable normal mode of meridionally concentrated gradients of the zonal-mean potential vorticity (PV). The flows are neither forced nor damped, except that moderate second-order horizontal diffusion is used to achieve an eddy-free state in a finite computational time. The final zonal-mean states are typically characterized by a well-defined barotropic jet that is not sufficiently stable in the sense of Charney and Stern but stable for all zonal wavenumbers allowed by the geometry of the channel. It is shown that vertical asymmetry in the meridional arrangement of PV leads to (a) production of barotropically sheared jet and (b) shift in the zonal scale of baroclinic instability and subsequent neutralization. It is argued that the extent of the meridional arrangement necessary to suppress the most momentum-transporting baroclinic wave determines the width of the adjusted flow. This width is roughly proportional to the initial zonal scale of the mode on the f plane but constrained by a beta-related critical mixing length on the beta plane.

Relationship to other theories (e.g., barotropic governor and geostrophic turbulence) is discussed, along with the relevance of the theory to the earth’s midlatitude troposphere.

Corresponding author address: Dr. Noboru Nakamura, Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis Avenue, Chicago, IL 60637.

Email: nnn@bethel.uchicago.edu

Abstract

Baroclinic adjustment hypothesis fails to account for the enhancement of the barotropic jet observed in idealized baroclinic-wave life-cycle simulations. In this paper, an adjustment theory more consistent with the numerical results is developed through a careful examination of life-cycle experiments and nonseparable eigenvalue problems using the two-layer model.

In all cases examined, nonlinear eddies emanate from an unstable normal mode of meridionally concentrated gradients of the zonal-mean potential vorticity (PV). The flows are neither forced nor damped, except that moderate second-order horizontal diffusion is used to achieve an eddy-free state in a finite computational time. The final zonal-mean states are typically characterized by a well-defined barotropic jet that is not sufficiently stable in the sense of Charney and Stern but stable for all zonal wavenumbers allowed by the geometry of the channel. It is shown that vertical asymmetry in the meridional arrangement of PV leads to (a) production of barotropically sheared jet and (b) shift in the zonal scale of baroclinic instability and subsequent neutralization. It is argued that the extent of the meridional arrangement necessary to suppress the most momentum-transporting baroclinic wave determines the width of the adjusted flow. This width is roughly proportional to the initial zonal scale of the mode on the f plane but constrained by a beta-related critical mixing length on the beta plane.

Relationship to other theories (e.g., barotropic governor and geostrophic turbulence) is discussed, along with the relevance of the theory to the earth’s midlatitude troposphere.

Corresponding author address: Dr. Noboru Nakamura, Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis Avenue, Chicago, IL 60637.

Email: nnn@bethel.uchicago.edu

1. Introduction

Transient baroclinic eddies are a hallmark of the earth’s midlatitude climate. Correct representation of their effects is essential for building any idealized climate model that does not resolve eddies explicitly. Among various proposals for such parameterization is the baroclinic adjustment hypothesis, which assumes that the eddies keep the zonal-mean state close to being neutral with respect to baroclinic instability (Smagorinsky 1963; Stone 1978). In the framework of quasigeostrophic theory, neutrality is achieved by eliminating the meridional gradients of potential temperature (or vertical shear) at the surface or those of the interior potential vorticity (PV) (Lindzen and Farrell 1980) and thereby satisfying the sufficient condition for stability due to Charney and Stern (1962, CS hereafter).

This conjecture is attractively simple but has been met with various challenges. First and utmost, the statistical mean state of a forced-dissipative atmosphere, whether observed or simulated, appears to be far from neutral. Cehelsky and Tung (1991) argue, in the context of a two-layer model, that neutrality is not enforced for all wavenumbers (as is guaranteed by the CS criterion) but only for the most energetic wave. In their calculation, the dominant wave has a much larger zonal scale than the most unstable wave due to upscale energy cascade, and the time-mean state is neutral to this dominant wave but remains supercritical to shorter waves. Stone and Branscome (1992) obtain similar results (see also Whitaker and Barcilon 1995; Welch and Tung 1998).

A more fundamental problem is that, even in the absence of forcing, damping, or upscale cascade, the baroclinic adjustment theory does not adequately describe the mean flow modification observed during idealized baroclinic-wave life-cycle simulations. For example, the normal mode with zonal wavenumber 6 or 7 on a sphere equilibrates by returning kinetic energy to the zonal-mean flow and creates a distinctive barotropic jet (Simmons and Hoskins 1978, 1980; Gutowski et al. 1989). A similar tendency is recognized in some observed life cycles in the Southern Hemisphere (e.g., Randel and Stanford 1985). Feldstein and Held (1989) demonstrate using the two-layer model that the “barotropitization” of the mean flow is inevitable even when supercriticality is weak as long as there are critical lines in the domain. On the contrary, baroclinic adjustment focuses on the destruction of vertical shear and says very little about the creation of horizontal shear.

More importantly, the life-cycle simulations do not necessarily produce a final mean state that satisfies the CS stability condition: the potential temperature gradients at the surface are modified but not destroyed globally, and the PV gradients near the tropopause are hardly diluted (see Fig. 12 of James 1987). This is contrary to the weakly nonlinear theory (Pedlosky 1987; Warn and Gauthier 1989) in which the waves do equilibrate by rendering the flow “CS-stable,” consistent with the adjustment hypothesis. James (1987) argues that the enhanced horizontal shear associated with the jet is responsible for the suppression of baroclinic instability (“barotropic governor”). One might hypothesize, therefore, that a barotropic jet is organized in such a way as to neutralize baroclinic eddies, and that the geometry of the adjusted flow is parametrically dependent on the known properties of the eddies. A quantitative theory based on this modified adjustment hypothesis is yet to be formulated, however, partly because there is no useful theorem that relates barotropic shear and neutrality and partly because the eddy momentum flux that drives barotropic jet is hard to parameterize. Some theoretical progress has been made through the WKB solutions of nonseparable eigenvalue problems (e.g., Ioannou and Lindzen 1986; Roe and Lindzen 1996), but the relevance of these studies to the adjustment process that entails strong flow curvature is debatable. A fuller understanding of nonlinear equilibration in the life-cycle simulations is necessary toward a more satisfactory theory of“baroclinic–barotropic” adjustment.

In this paper, we investigate the determinism of the meridional scale of barotropic jets that emerge from life cycles of baroclinic waves. To develop a theory in as simple a setting as possible, we consider flow evolutions in the quasigeostrophic two-layer model. In our experiments flows are initially strongly supercritical within a narrow latitude band. Eddies arising from the unstable normal mode of such flows are therefore meridionally localized but strongly nonlinear, and they tend to alter the meridional scale and structure of the flow as they equilibrate. As we shall see, the final zonal-mean state is generally characterized by a well-defined barotropic jet that is not CS-stable but stable to all wavenumbers allowed by the geometry of the model.

The adjustment process will be interpreted as “meridional rearrangement,” rather than destruction, of the zonal-mean PV gradients by baroclinic eddies. An important point is that this rearrangement is asymmetrical between the two layers. This causes (a) production of barotropically sheared jet through PV inversion and (b) shift in the zonal scale of baroclinic instability that eventually leads to neutralization via the wavenumber constraints (i.e., short- and long-wave cutoffs). It is proposed that the extent of the meridional arrangement necessary to suppress the most momentum-transporting baroclinic wave determines the width of the barotropic jet. A simple analytic model (Nakamura 1993a) will be deployed to substantiate these points.

The next section outlines the model architecture and solution procedures. In section 3 the results of the nonlinear calculation on the f plane are described, followed by the beta-plane results in section 4. Since the time dependence of the solutions share many common features with previous results (e.g., Feldstein and Held 1989; Nakamura 1993b; Frisius 1998), we will focus on the stability of the equilibrated flows. Much of the theoretical development takes place in section 5 with the aid of the analytic model. Implications to the earth’s midlatitude jet and the relationship to previous results are discussed in the concluding section.

2. The two-layer model

The primary numerical tool in this study is the quasigeostrophic two-layer model with an equal mean layer thickness. In each layer, the eddy PV (q′) and the meridional gradient of the zonal-mean PV (Q) are governed by
i1520-0469-56-13-2246-e2-1a
and
i1520-0469-56-13-2246-e2-2a
Here the subscripts 1 and 2 denote the upper and lower layers, respectively, and the overbar indicates the zonal average ψ′ and U are the eddy geostrophic streamfunction and zonal-mean velocity, respectively. The constant LR is the internal Rossby radius and ∇2 is the horizontal Laplacian. The only nonconservative term is the second-order horizontal diffusion applied to the eddy PV (there is no thermal relaxation or Ekman damping). Compared with the more popular hyperdiffusion acting on relative vorticity, this less scale-selective viscosity allows an eddy-free final state to emerge in a relatively short computational time. Meanwhile, the gradient of the zonal-mean PV is undiffused so that the effect of artificial dissipation on the adjusted flow is minimized. In addition to the horizontal diffusion, a scale-independent linear damping is applied to (2.1a) and (2.2a) in the sponge regions near the zonal boundaries to suppress spurious reflection of the Rossby waves (only for the beta plane). The e-fold damping rate is βLR at the boundaries and linearly decreases to zero at the distance 2LR from the boundaries. Specific values of the parameters used for each experiment are listed, in dimensional forms, in appendix A.
Equations (2.1a) and (2.2a) are solved for a zonally periodic channel using the pseudospectral transform in the zonal direction and finite differencing in the meridional direction. At each time step ψn and Un are inverted from qn and ∂Qn/∂y, respectively, with the boundary conditions
ψnUnyLy
This condition assumes that the zonal boundaries are sufficiently distant so the flow remains quiescent there. We also assume that barotropic shear ∂(U1 + U2)/∂y is either zero or symmetric at the two boundaries. Therefore, by integrating (2.2b) meridionally and summing over both layers,
i1520-0469-56-13-2246-e2-4
This serves as an integral constraint for ∂Qn/∂y to be specified. In all cases examined, Ly = 20LR is assumed. We retain 400 grid spacings between the two zonal boundaries and 32 zonal harmonics. The specific forms of the initial conditions will be described later.
A nonseparable eigenvalue problem will be solved to examine the stability of the adjusted zonal-mean flow. It is formed by neglecting the quadratic terms in (2.1a). Assuming the normal mode form
ψnnyikxctn
(2.1a) becomes
i1520-0469-56-13-2246-e2-6
Note that we retain the same diffusion term as in the initial-value problem. The linear damping associated with the sponge layers hardly affects the solution since the mode amplitudes are negligible there in all cases examined. We discretize (2.6) in y by finite differencing with 100 grid spacings between the two zonal boundaries (a quarter-resolution of the initial-value problem). After some arrangements, this yields a generalized eigenvalue problem Az = cBz, where A and B are 200 × 200 matrices. A standard QR algorithm (LAPACK; http://www.netlib.org/lapack) is used to obtain the eigenvalue c and eigenvectors z.

To shed light on the results of the initial-value problem analytically, we will introduce in section 5 a further idealization to (2.6) by assuming that PV is piecewise constant. This allows us to analytically examine the behavior of shortwave cutoff in response to the asymmetrical arrangement of PV “jumps.” The general formulation of this analytic model is outlined in Nakamura (1993a) and thus will not be repeated here. More details will be given in sections 5b and 5d.

3. The f-plane results

We consider an initial mean flow given by the meridional PV gradients of the form
i1520-0469-56-13-2246-e3-1
wherein L1 = L2 = 1.5LR, y1 = −y2 = 0.05LR, and Γ1 = −Γ2, which satisfies (2.4) (β = 0). The zonal-mean flow Un is numerically inverted from (3.1) using (2.2b). Both ∂Qn/∂y and Un are isolated from the zonal boundaries, and their sign is opposite between the two layers (Figs. 1a,b). Hence, the flow is baroclinically unstable. A slight asymmetry is introduced through y1 and y2, the significance of which will become evident later. The linear stability of this flow is shown in Fig. 1c. The most unstable mode (S1) is used to perturb the zonal-mean state, with the channel length adjusted to one wavelength.

The model is integrated in time until eddy amplitude decays away and the zonal-mean flow reaches a steady state. The three lower rows of Fig. 1 show the meridional profiles of the equilibrated state and its stability for three zonal wavenumbers of the initial mode k = 1.3, 1.0, and 0.8 (normalized by L−1R). Although the modes grow at different rates, all cases equilibrate in a similar way: baroclinic eddies decay by returning kinetic energy to the zonal-mean flow, giving rise to a prograde–retrograde jet pair that is sheared more horizontally than vertically and somewhat wider than the initial baroclinic zone. Much of the mean flow modification is achieved by the primary life cycle, although weak secondary instabilities keep the flow unsteady for some time. The scale of the produced jet increases as the channel length (=2πk−1) is increased: while k = 1.3 (channel length = 4.8LR) results in a shear layer that spans ∼10LR, k = 0.8 (7.8LR) expand it to ∼26LR (Figs. 1d,g,j). The jets become more barotropic as they widen, but the maximum velocity does not vary very much. The PV gradients show variation at scales much smaller than the flow, particularly for wide jets (Figs. 1e,h,k). In all cases the equilibrated PV gradients include both signs in each layer, so the adjusted flow is not sufficiently stable in the sense of CS.

The right column of Fig. 1 reveals that in the equilibrated state instabilities are cut off at the wavelength exactly equal to the length of the channel, indicated by the arrows.1 The waves longer than the channel length remain unstable, but they cannot be realized. As the flow widens, the stationary mode (A) that strides over both jets recedes to a greater zonal length, and the growth rates are dominated by a pair of counterpropagating instabilities (B) that live on either of the jets (Figs. 1f,i,l). Although the sense of horizontal tilt of the initial mode S1 is such as to sustain countergradient eddy momentum flux, that of modes A and B is the opposite. This suggests that both are close relatives of barotropic instability, consistent with the double-signed PV gradients in each layer.

Figure 2 is the same as Fig. 1 but for an initial flow with a slight vertical asymmetry in the concentration of the PV gradients [(3.1) with y1 = −y2 = 0, L1 = 0.9LR, L2 = 1.1LR, Γ1 = −1.22Γ2]. The equilibrated flow exhibits a prograde jet at the channel center and two retrograde jets at the flanks (Figs. 2d,g,j). Despite the sensitivity of the flow geometry to the initial condition, the tendency to produce barotropic jet and the dependence of the meridional scale of the equilibrated flow on the zonal length of the channel are robust: the two scales are again approximately proportional, and the flows are adjusted such that the shortwave cutoff occurs precisely at the length of the channel (Figs. 2f,i,l). None of the flows satisfy the CS stability.

In the above two experiments, the vertical asymmetry fed in the initial zonal-mean flow is crucial, no matter how small it is. As Nakamura (1993b) points out, vertically antisymmetric (purely baroclinic) zonal-mean flows preserve the symmetry on the quasigeostrophic f plane and hence preclude barotropic component for all time, even in the presence of strong nonlinearity. Thus, when the vertical asymmetry is removed from the two cases (y1 = y2 = 0, L1 = L2 = 1.5LR, Γ1 = −Γ2), a very distinct zonal-mean state ensues despite the similar initial condition (Figs. 3a–c): two well-defined PV “edges,” each accompanied by jets, separate sideways. Figures 3d and 3e show an intermediary state generated by k = 1. The region between the edges is filled with eddies (not shown). The failure to equilibrate is consistent with the stability of the intermediary state. The growth rates of S1 somewhat diminish with time, but the unstable range of zonal wavenumbers does not change appreciably (Figs. 3c,f). Thus the flow remains unstable at k = 1. The instability keeps entraining mass into the eddy-active region, pushing the PV edges into the quiescent region in a manner described by Nakamura (1996, his Fig. 7a). Once the edges reach the boundaries, eddies fill the entire domain and PV is completely homogenized. At this point the flow is rendered CS-stable and all available potential energy is exhausted.

This solution is nevertheless very special and, in fact, structurally unstable, in that even a small asymmetry in the initial mean flow leads to a runaway production of barotropic jet as shown above and by Nakamura (1993b).

4. The beta-plane results

Now the beta effect is introduced while the same zonal-mean flow as in Fig. 3a is used to initialize the model. The magnitude of beta is only 20% of the peak PV gradient at the center of the flow, so the initial flow is still locally strongly supercritical. Even through the mean flow is vertically antisymmetric, the beta effect breaks the vertical symmetry in PV and gives rise to barotropic process in the subsequent evolution. The geometry and stability of the initial flow are depicted in Figs. 4a–c.

The adjusted zonal-mean states and their stability properties are summarized in the rest of Fig. 4. Unlike the symmetry-preserving f-plane case, the zonal-mean flow generally develops meridionally confined barotropic jets, reminiscent of the f-plane results in Fig. 2d. The acceleration of the flow at the center of the channel and deceleration at the flanks are consistent with the previous wave activity diagnostics (Edmon et al. 1980;Magnusdottir and Haynes 1996). In the upper layer, the rearrangement of the PV gradients is most pronounced at the flanks of the prograde jet, the sites for wave breaking and subsequent secondary instabilities (Haynes 1985). Diminished (even slightly negative) PV gradients appear just outside of the core of the prograde jet, while the PV gradients are reinforced at the axis of the jet (Nakamura 1996, Fig. 8a). In the lower layer, mixing tends to weaken both negative and positive gradients (Figs. 4e,h,k,n). As a result, the end states are much less supercritical than the corresponding initial states. This is reflected in the much reduced growth rates, particularly for the two long waves (Figs. 4l,o). Nevertheless, none of the end states are CS-stable quite yet, and the wavenumber quantization is still effective as a mechanism of equilibration. Unlike the f-plane cases, the scale shift of the growth rates shows a more complicated pattern. While k = 1.3 and 1.0 produce the shortwave cutoff at the channel length similar to the f-plane cases (Figs. 4f,i), k = 0.75 and 0.63 result in the growth rates that occupy somewhat shorter zonal scales. The end state of k = 0.75 actually falls on a valley between the two unstable modes—a rather subtle neutrality (Fig. 4l). The end state of k = 0.63 is completely stable at the channel length but marginally stable to the second zonal harmonic (Fig. 4o).

Perhaps the most striking feature of Fig. 4 is that the meridional extent of the adjusted flow is far less sensitive to the zonal scale of the flow than the f-plane cases (e.g., Fig. 2). On the other hand, the flow scale does depend on beta: when the value of beta is reduced to half (i.e., the initial supercriticality being twice as large), the meridional extent of the adjusted flow increases significantly, although much of the expansion is in the width of the retrograde flow outside the core of the prograde jet (Fig. 5 for k = 0.75, to be compared with Figs. 4j and 4k). The role of beta in determining the meridional extent of the adjusted flow will be discussed further in the next section.

5. Some analytical insights on the numerical results

A coherent picture that emerges from the preceding numerical results is that meridionally confined, strongly supercritical flows find it easier to neutralize themselves via the wavenumber constraints than by eliminating instability altogether. The required shift in the zonal scale of instability occurs when the meridional arrangement of the zonal-mean PV is vertically asymmetric. In this section, we present some analytical results to shed light on the mechanics of the scale shift.

a. Invariance of the cutoff scales in the purely baroclinic flows on the f plane

The vertically antisymmetric flows on the f plane do not equilibrate via shortwave cutoff, as shown in Fig. 3. To see why this is the case, assume ν = β = 0 and U1(y) = −U2(y) ≡ ΔU(y) in (2.6). It is then easy to find the following particular solutions:2
i1520-0469-56-13-2246-e5-1a
The vanishing c means that these modes are steered at the midlevel (thus stationary) and have zero growth rates. Since all growing normal modes under consideration are stationary owing to the flow symmetry, (5.1a) and (5.1b) are two of the marginal stability solutions (i.e., cutoffs). For example, when ΔU is constant with y, (5.1a) and (5.1b) represent the longwave and shortwave cutoffs, respectively, of the transverse mode [the one with zero meridional wavenumber; Pedlosky (1987), section 7.11]. More generally, (5.1) states that there is a mode whose meridional structure is identical to that of the flow at the marginal stability [as long as ΔU satisfies the same boundary conditions as the mode amplitudes; in our case both are vanishing: (2.3)]. For a meridionally symmetric ΔU that has no zeros, this pertains to the first symmetric mode—the one (usually) with the greatest growth rate.

It is important to note that the cutoff wavenumbers in (5.1) are independent of the meridional profile of ΔU. It implies that if the flow is initially unstable to the first symmetric mode, it remains unstable for all time because the flows under consideration preserve the symmetries (Nakamura 1993b) and hence the cutoff scales of the mode. This is consistent with Figs. 5e and 5f (the slight deviation of the longwave cutoff from k = 0 is apparently due to a numerical diffusion) and shows that the vertically symmetric arrangements of PV cannot neutralize the flow via the wavenumber quantization.

b. A simple model of asymmetric arrangement of PV

To illustrate how the asymmetric arrangement of the PV gradients gives rise to barotropic jet and affects the cutoff scales at the same time, we consider an idealization to (2.6) wherein PV in each layer is piecewise constant. Here we also assume that the domain is meridionally infinite and beta is zero. A detailed formulation of the model is discussed in Nakamura (1993a). Specifically, consider an arrangement of three PV jumps as in Fig. 6a. The sign of the jumps is opposite between the two layers so the flow is baroclinically unstable, but the two negative jumps in the lower layer are meridionally separated by a distance 2L. This PV configuration is somewhat inspired by Fig. 2e but not intended to be a full representation of the nonlinear result. (For example, the secondary barotropic instability will not be resolved, since the PV jumps are of single sign in each layer.) The following flow corresponds to this PV configuration (Fig. 6b):
i1520-0469-56-13-2246-e5-2a
Note that the barotropic component of the flow (UB) is an isolated point jet. The flow becomes purely baroclinic (vertically antisymmetric) when L = 0. Following Nakamura (1993a), the modes with zero perturbation PV are sought in each layer segment and then matched at the PV discontinuities. In doing so, only the meridionally symmetric modes are considered since the antisymmetric modes cannot be unstable in this model. This leads to a quadratic dispersion relation listed in appendix B.

The growth rates are plotted in Fig. 6c as a function of wavenumber and separation L. As L increases, the shortwave cutoff shifts to longer zonal scales. This is because a greater meridional separation requires a greater amplitude penetration of the edge waves to enable an unstable interaction between them (just as in Eady’s problem). Since the penetration scale of an edge wave is proportional to the zonal wavelength, the spectrum of instability shifts to a longer range.

If L is taken as a measure of the meridional arrangement (mixing) necessary to neutralize a purely baroclinic flow with respect to baroclinic instability, its value is determined by the shortwave cutoff for the given zonal wavenumber. For k = 1, L ≈ 1.5, for example. This L also gives a meridional scale of the barotropic jet in the adjusted flow. From Fig. 6c it is clear that the shortwave cutoff of a longer wave requires a greater L. This is consistent with the f-plane numerical results (Figs. 1, 2).

c. Barotropic adjustment

In the nonlinear calculations of sections 3 and 4, PV gradients of opposite sign, often with a comparable magnitude, appear within the same layer in the wake of the primary baroclinic instability. This gives rise to secondary barotropic instabilities (Haynes 1985; Frisius 1998). However, it is unlikely that the secondary instabilities alter the adjustment theory significantly. Consider, for example, the same point jet as used for UB in (5.2b) but placed in a single layer (Figs. 6d,e). This jet is barotropically unstable. The growth rates of the unstable modes are depicted in Fig. 6f. The zonal scale of the shortwave cutoff is approximately proportional to the meridional separation. The meridional separation required to suppress instability via the shortwave cutoff is comparable to that for baroclinic instability (Fig. 6c) for long wavelengths (k < 1).

d. Beta effect

On the beta plane, meridionally confined negative zonal-mean PV gradients are initially surrounded by a positive PV gradient (beta) in the lower layer. This means that a local minimum in PV appears just north of a local maximum. As mixing homogenizes these local extrema, the negative PV gradients diminish and eventually disappear. This causes the supercriticality to cease. If the negative PV gradients are initially concentrated to within −δLyδL, then the meridional scale over which PV can be homogenized (Lh) is approximately given by
i1520-0469-56-13-2246-e5-3
Hence, the meridional extent of the adjustment by strongly nonlinear (i.e., PV mixing) baroclinic instability is likely limited by this critical mixing length beyond which the flow becomes subcritical. Although no beta-plane experiments in section 4 actually achieve subcriticality (i.e., CS stability), the supercriticality does diminish and the meridional extent of the adjusted flow is rather insensitive to the zonal scale of the channel (Fig. 4). This suggests that the final flow scale is indeed constrained by (5.3).

It is noteworthy that (5.3) is essentially similar to the theoretical estimate for the thickness of nonlinear critical layer (Killworth and McIntyre 1985). If we view the retrograde flows emerging at the flanks of the prograde jet in Figs. 4 and 5 as the result of wave activity dissipation in the nonlinear critical layers, the expansion of this retrograde flow in Fig. 5 is consistent with the expansion of nonlinear critical layers due to the reduced beta.

The essence of this additional physics can be incorporated in the following variation of the above analytical model (Figs. 6g,h). The PV configuration we consider is similar to Fig. 6a, but the amount of the negative PV jump in the lower layer diminishes with increasing L as −ε + βL. At L = ε/β, the PV jumps change sign and the flow becomes CS-stable. This critical separation is analogous to (5.3). The arrangement of the lower-layer PV in this model is sketched in Fig. 7. The mean wind that corresponds to this PV geometry (Fig. 6h) is given by, instead of (5.2b),
i1520-0469-56-13-2246-e5-4
The flow reduces to (5.2) when β = 0. An ad hoc aspect of (5.4) is that the beta-plane approximation is employed only inside the shear layer (−LyL). The f plane is used in the outer regions to keep the barotropic flow bounded. The stability analysis leads to a quadratic dispersion relation for the meridionally symmetric modes (appendix B). The growth rates are plotted in Fig. 6i for β = 0.5ε/LR as a function of the zonal wavenumber and meridional separation. The difference from Fig. 6c (β = 0) is evident: all modes are stable for L > ε/β as explained above. Furthermore, for an intermediate separation 0 < L < ε/β, there appears a longwave cutoff. Compared with Fig. 6c, the long waves are neutralized at a much smaller separation. This quantitatively captures the behavior of the beta-plane calculations of section 4.

6. Discussion

As pointed out in the introduction, failure to predict the emergence of the barotropic jet, which is ubiquitous in the simulated and observed baroclinic wave life cycles, constitutes a fundamental flaw in the baroclinic adjustment hypothesis. This is in a sense less excusable than the apparent lack of its evidence in the forced-dissipative environment (e.g., Vallis 1988), which may be attributed to inefficiency, rather than irrelevance, of the adjustment against other competing processes.

We have shown that barotropitization can be adequately represented, at least in the two-layer model, if we allow vertical asymmetry in the meridional rearrangement of the zonal-mean PV gradients. Idealized life cycle calculations confirm that strongly supercritical PV gradients, initially confined to within a narrow latitude band in a wide domain, are rearranged by baroclinic eddies into a somewhat broader but more vertically asymmetric state accompanied by a well-defined barotropic jet. The vertical asymmetry may grow as a result of asymmetric mode structure (Nakamura 1993b), or by meridional emanation of Rossby waves and their encounter with critical lines, which is pronounced only in the upper layer (Edmon et al. 1980; Feldstein and Held 1989; Magnusdottir and Haynes 1996). Regardless of the driving physics, the “asymmetric broadening” of the PV gradient zone leads to a shift in the zonal scale of instability, which eventually causes the flow to equilibrate via the wavenumber constraints (i.e., shortwave and longwave cutoffs) without satisfying the CS stability condition. It is argued that the extent of the meridional arrangement necessary to suppress the most momentum-transporting wave determines the meridional scale of the final flow. On the f plane, this scale is approximately proportional to the zonal scale of the initial perturbation (=the length of the channel), although the profile of the adjusted flow depends sensitively on the initial asymmetry. On the beta plane, the meridional extent of arrangement is constrained by the critical mixing length given by (5.3). As a result, the long waves, which would modify the flow to a great meridional extent on the f plane, are unable to do so and render the flow nearly (if not precisely) CS-stable.

The above equilibration mechanism is fundamentally different from the weakly nonlinear theory in which the scales of the flow are fixed while the PV gradients are destroyed. Our results are more analogous to the adjustment theory for the tropopause height (Lindzen 1993). Using the neutrality condition for Eady’s (1949) model on the f-plane channel with the meridional width L,
i1520-0469-56-13-2246-e6-1
Lindzen (1993) argues that the tropopause height H is adjusted toward the baroclinic neutrality for the given values of kc (zonal wavenumber), N (Brunt–Väisälä frequency), and L. Although Thuburn and Craig (1997) discount such adjustment in a general circulation model, nonquasigeostrophic two-dimensional (xz) life cycles of Eady waves (Nakamura and Held 1989; Garner et al. 1992; Nakamura 1994) provide a numerical (albeit exotic) paradigm in which equilibration does occur in this fashion: an unstable normal mode with a zonal wavenumber k grows until mixing increases NH to the point k = kc, when the flow is neutralized (for this wavenumber). In these experiments, the meridional potential temperature gradient is fixed, so the CS sufficient condition for stability is never met.

Lindzen (1993) assumes, based on the WKB analysis, that L in (6.1) represents the width of the jet. Then, can (6.1) (or its two-layer counterpart on the f plane) be used to predict the width of the equilibrated flow for the given kc, N, and H? Our results send a caveat against such an attempt: (6.1) gives greater cutoff wavenumbers for wider jets, which is the opposite of our f-plane results. Clearly (6.1), based on unsheared or slowly varying channel geometry, cannot be used to predict the meridional scale of the neutralized flow, which is far removed from the basic state to which the relation is valid.

The PV-based view of the present study de-emphasizes the role played by horizontal shear as a suppressor of baroclinic instability (James 1987). James views horizontal shear as an external agent: even constant shear with no contribution to the PV gradients suppresses instability. In our problem, a constant horizontal shear cannot be realized due to the imposed boundary conditions (vanishing winds) so the flow is completely determined by the internal PV gradients. A locally large barotropic shear still emerges, but only in response to the vertically asymmetric arrangement of the PV gradients. In this sense horizontal shear can be viewed as a by-product, rather than a governor, of equilibration.

In this paper it is assumed that baroclinically unstable zonal-mean PV gradients are fed at a small meridional scale. This is in contrast with the geostrophic turbulence regime (Rhines 1975, 1979; Panetta and Held 1988; Panetta 1993) wherein narrow jets emerge from broad PV gradients. In reality, neither regime truly captures the earth’s midlatitudes wherein there is not much separation between the radiative forcing scale and the scale of the jet. Given the different determinism of the jet scale, it is perhaps fortuitous that both theories predict a similar jet scale for a parameter range typical of the earth’s midlatitudes (on the beta plane).

We have considered life cycles in the two-layer model with single zonal wavenumber and its subharmonics to isolate the dynamics of barotropic jet formation. The long list of neglected physics makes it hard to generalize the results for the real atmosphere. In a forced-dissipative environment with a full spectrum of wavenumbers, the adjustment theory presented here can fail as a model of climate for the same reason that the original baroclinic adjustment hypothesis and Lindzen’s (1993) tropopause adjustment theory fail: competition against forcing and inability to choose a priori the single most relevant wavenumber. Still, we deem it important for an adjustment hypothesis to be consistent with what baroclinic eddies attempt to do in an idealized environment. In this regard the present study improves upon the original baroclinic adjustment theory, which excludes the adjustment to horizontal shear, providing constraints on the scale of the jet at least as an asymptotic limit.

In a separate paper (Song and Nakamura 1999) stability analyses of isolated jets are performed based on the semigeostrophic Eady model on a semi-infinite domain. Unlike the quasigeostrophic model, the semigeostrophic model allows concomitant adjustments to static stability and tropopause height. Barotropic jet formation and scale shift in baroclinic instability, similar to the f-plane results of the present study, are observed when potential temperature is arranged asymmetrically at the ground and tropopause. This suggests that the effects of stratification is inconsequential and that the two-layer model captures the essence of the baroclinic–barotropic adjustment. It has been brought to our attention that (5.1b) has been derived previously by Pedlosky and Klein (1991) and, for a more special case, by Hart (1990).

Acknowledgments

The authors thank three anonymous reviewers for constructive criticisms, and R. Lindzen for useful comments on the earlier version of the manuscript. This research is supported by NSF Grant ATM9627040.

REFERENCES

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  • ——, and B. F. Farrell, 1980: The role of polar regions in global climate, and a new parameterization of global heat transport. Mon. Wea. Rev.,108, 2064–2079.

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  • ——, 1993b: Momentum flux, flow symmetry, and the nonlinear barotropic governor. J. Atmos. Sci.,50, 2159–2179.

  • ——, 1994: Nonlinear equilibration of two-dimensional Eady waves:Simulations with viscous geostrophic momentum equations. J. Atmos. Sci.,51, 1023–1035.

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APPENDIX A

Model Parameters

The following table lists the values of the parameters, in dimensional forms (SI unit), used for most of the initial-value problems discussed in this paper. See sections 2 and 3 for the definition of the parameters. Here LR = 106 m and Ly = 20LR for all cases.

APPENDIX B

Dispersion Relation for the PiecewiseConstant PV Model

The dispersion relation for the meridionally symmetric baroclinic waves discussed in section 5 is given by
c2AcB
where
i1520-0469-56-13-2246-eb-2
When β = 0 the above gives the dispersion relation for the flow given by (5.2), otherwise by (5.4).

Fig. 1.
Fig. 1.

Meridional profiles of the zonal-mean states and their stability characteristics associated with an f-plane life cycle experiment. (a)–(c) initial condition; (d)–(f) final state for k = 1.3; (g)–(i) final state for k = 1.0; (j)–(l) final state for k = 0.8. The zonal wavenumber k is normalized by the inverse Rossby radius L−1R. (left) The zonal-mean flow as a function of y (in LR); the lower-layer profile is dashed. Unit is 0.2Γ1 L2R (=10 m s−1). (middle) Same as the left column but for the zonal-mean PV gradients. Unit is Γ1 (=5 × 10−11 s−1 m−1). (right) Growth rates of the unstable modes for the mean states shown in the two left columns as a function of k. Unit is 0.2 Γ1 LR (=10−6 s−1). The arrows indicate the wavenumber corresponding to the length of the channel. S1, first meridionally symmetric mode (slightly tilted due to the flow asymmetry); A1, first antisymmetric mode; S2, second symmetric mode; A, stationary mode whose amplitude maximizes at both jet axes; B, two counterpropagating modes whose amplitude maximizes at either one of the jet axes; C, two counterpropagating modes whose structure is nearly antisymmetric about each jet axis.

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 2.
Fig. 2.

Same as Fig. 1 except for the meridionally symmetric initial condition. S1, first symmetric mode; S2, second symmetric mode; A1, first antisymmetric mode. (i) and (l) Modes with similar growth rates correspond to various combinations of meridional symmetries [i.e., symmetry and antisymmetry about the three axes of the jet in (g) and (j)].

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 3.
Fig. 3.

Same as Fig. 2 but for a vertically antisymmetric initial condition. The zonal-mean states in the bottom row are the time average between t = 210 and 240 (t is normalized by Γ−11L−1R = 2 × 104 s), and the growth rates in (f) pertain to this averaged state. S1, first symmetric mode; S2, second symmetric mode; A1, first antisymmetric mode.

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 4.
Fig. 4.

Same as Fig. 2 but for the beta-plane experiment. (top) Initial condition; (row 2) k = 1.3; (row 3) k = 1.0; (row 4) k = 0.75; (bottom) k = 0.63. (f), (i), (l), (o) Arrows labeled “1” indicate the zonal wavenumber corresponding to the channel length, whereas those labeled “2” indicate the wavenumber of the second zonal harmonic. S1, S2, and S3 are meridionally symmetric modes and A1 is the antisymmetric mode.

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 5.
Fig. 5.

Same as Figs. 4j and 4k but for half the value of beta.

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 6
Fig. 6

(a) A diagram showing the meridional arrangement of three PV discontinuities in two layers. (b) Zonal-mean flow that sustains the PV geometry shown in (a). Here L = 1.2LR is assumed. Solid curve, upper-layer flow; dashed curve, lower-layer flow; thin solid curve, average of the upper- and lower-layer flows. (c) Growth rates of the unstable mode for the above flow as a function of the zonal wavenumber and meridional separation. (d) Same as (a) but for a barotropically unstable situation in a single layer. (e) Zonal-mean flow that sustains the PV geometry shown in (d). (f) Same as (c) but for the flow defined by (d) and (e). (g) Same as (a) but with a beta effect. See text for details. (h) Zonal-mean flow for (g). (i) Same as (c) but for the flow defined by (g) and (h).

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Fig. 7.
Fig. 7.

A schematic of the PV mixing in the lower layer (beta plane). The thick solid lines indicate PV as a function of y. (a) The negative PV gradients (a jump) are meridionally isolated initially. (b) As PV is partially mixed, the negative PV jump at the edges diminishes in magnitude. (c) When the critical mixing length is reached, the negative PV jumps disappear and the flow becomes baroclinically neutral. Notice that in (5.4), the outer regions with the constant PV gradient are replaced by the ones with constant PV (dashed lines).

Citation: Journal of the Atmospheric Sciences 56, 13; 10.1175/1520-0469(1999)056<2246:BBAIAM>2.0.CO;2

Table A1. List of the parameters used for the initial-value problems in this paper. The values are in dimensional forms (SI unit).

i1520-0469-56-13-2246-t01

1

In getting this result, it is important to use the same diffusion coefficient in the stability analysis as is used in the intial-value problem. In most cases examined in this paper, the adjusted flow supports weak shortwave instabilities in the inviscid limit beyond the shortwave cutoff. These instabilities are readily suppressed by the diffusion used in the numerical model.

2

See note added in proof at end of section 6.

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