1. Introduction

From theoretical (Rhines 1975) and numerical (Williams 1978) studies, it is well known that in the presence of the meridional planetary vorticity gradient β, an upscale energy cascade generates zonal jets in two-dimensional turbulence. Numerical simulations of quasigeostrophic (QG) turbulence show that if the width of the baroclinically unstable region of the initial flow is greater than the inverse of Rhines’s cascade-arrest scale, (β/2u_rms)^1/2, multiple zonal jets can emerge (Williams 1979, 1988) and persist for a remarkably long time (Panetta 1993). Because Panetta (1993) used a model that includes a full zonal and meridional spectrum, this tremendous persistence of the multiple zonal jets seems robust and not an artifact of the model resolution. Such persistent zonal jets can be relevant for several geophysical fluids, including the giant planetary atmospheres (Williams 1978, 1979) and the ocean (Treguier and Panetta 1994).

While the dynamics of multiple zonal jets is of great interest, many of the fundamental properties of the multiple zonal jets are not well understood. For example, why do the multiple zonal jets found by Panetta (1993) persist with no sign of termination? Is the dynamics of each jet within a multiple jet flow different from that of a flow with a single jet? How would the eddy energy and eddy fluxes, whose parameterization has been a long-standing challenge, behave in inhomogeneous turbulence with such zonal jets?

In addressing the questions raised above, this paper describes some aspects of eddy fluxes and eddy energy that are associated with multiple zonal jets, focusing on the transition from a single jet to a double jet state. This paper is organized as follows. Section 2 briefly describes the model, and section 3 presents an overview of the results. Sections 4 and 5 describe the role of critical latitudes and unstable normal modes in the organization and maintenance of the multiple jets. The concluding remarks follow in section 6.

2. The two-layer quasigeostrophic model

The dimensionless equations for the two-layer QG model on a β-plane are

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and j = 1 and 2 refer to the upper and lower layers, respectively. The wind field is determined by the relation (u_j, υ_j) = (−∂ψ_j/∂y, ∂ψ_j/∂x). The horizontal length scale is the radius of deformation, λ. Time is nondimensionalized by λ/U₀, where U₀ is the horizontal velocity scale. The Ekman damping coefficient κ_M is included in the lower layer only, and δ_j2 represents the Kronecker delta. The additional parameters are κ_T (strength of radiative damping), ν (biharmonic diffusivity), and β (gradient of the Coriolis parameter). The values of κ_T and ν are fixed at 30⁻¹ and 6 × 10⁻³, respectively. A complete description of the model appears in Lee and Held (1991).

We use the following form for the initial zonal-mean zonal wind profile to balance the “radiative equilibrium temperature” field τ_e(y):

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This radiative equilibrium zonal wind field is constant in the center of the domain, and exponentially decays to zero outside of this central region. The initial upper and lower layer zonal mean zonal winds are set equal to U_e and zero, respectively (for example, see side panels in Figs. 1a and 1b). The same zonal wind profile was used by Panetta and Held (1988) and Lee and Feldstein (1996). The value of σ is fixed at 4, but W_c is varied. The parameter 2W_c measures the meridional width of the baroclinically unstable region of the initial flow.

The model uses finite differencing in the meridional direction and spectral transforms in the zonal direction. The channel width W is fixed at 90, where the walls are located at y = 0 and y = W. There are 300 grid points between the channel walls. The fundamental zonal wavenumber is 0.1, and there are 20 zonal waves in this model.

There are five sets of experiments; for each set, the values of β and κ_M are fixed, but the value of W_c is varied. One of the experimental sets represents a control, and the remaining four sets are designed to address the sensitivity of the results to the values of β and κ_M. Table 1 summarizes the five experimental sets and provides a title for each set and a range of values for W_c. Results from 60 different runs are presented, and each run is integrated for 2000 model days (λ/U₀) or more to ensure that a statistically steady state is obtained.

In this paper, the eddy energy ϵ (the sum of eddy kinetic and eddy available potential energy) is defined as the eddy energy averaged over the entire channel length in the zonal direction, but only over the “flat-top” region of width 2W_c in the meridional direction and over both layers. The precise definition of the eddy energy can be found in (3) and (4) below. Various terms in the energy budget, such as the baroclinic and barotropic eddy energy conversions, are also averaged in the same manner.¹

3. An overview

This section outlines the results from the five sets of experiments. We first describe results from the control run. Figures 1a and 1b show time–latitude segments of the vertically averaged zonal mean zonal winds for a disorganized, one-jet state and a well-organized, two-jet state, respectively. The value of W_c is 10 for the former case and 11 for the latter case. Comparing these two cases, it is clear that the two jets in the W_c = 11 case are remarkably persistent. This comparison also illustrates how dramatically the jet structure can change as W_c is varied by a small amount. This type of persistent, well-organized multiple jet is described in great detail by Panetta (1993), using a doubly periodic two-layer QG model.

Figures 2a and 2b show the meridional structure of the first empirical orthogonal function (EOF1, hereafter) of the zonal wind for W_c = 10 and W_c = 11 of the control experiment, respectively, along with the time-mean zonal-mean zonal winds. In both cases, the structure of EOF1 indicates that the dominant zonal jet variability is characterized by meridional displacement centered around the position of the time-mean jet, but the meridional displacement is more pronounced for W_c = 10, consistent with the time–latitude diagram (Fig. 1a). The mechanism for these meridional jet displacements is described in Lee and Feldstein (1996). For W_c = 11, the structure of EOF1 represents not only meridional displacements, but also a pulsation of the zonal wind at the channel center; when the two jets move toward (away from) each other, the strength of the interjet minimum at the channel center increases (decreases).

In every experimental set, the transition from one to two persistent jets does not occur abruptly at a precise value of W_c. Instead, there are ranges of values of W_c over which one and two jet states alternate in time. For example, such alternating flows are found for W_c = 10.5 in the control experiment. However, for a smaller value of κ_M (WD) or a larger value of β (HB), this transition occurs over a smaller range of values of W_c. Therefore, although the transition does not occur at a single value of W_c, we refer to the values of W_c over which the transition occurs as the “transition point.” Also, the values of W_c larger (smaller) than the transition point are referred to as above (below) the transition point. For example, in the control experiment, W_c = 10 is below the transition point, while W_c = 11 is above the transition point.

When the transition from one to two jets takes place, ϵ decreases (see Fig. 3a), reversing the tendency of an increase in ϵ with an increase in W_c. For example, the value of ϵ for W_c = 11 is smaller than that for W_c = 10. As W_c is further increased toward W_c = 19, ϵ also increases and a two jet state is maintained. At W_c = 20, a three-jet state is established and the value of ϵ drops once again. Similar behavior is found when a four-jet state is established at W_c = 28.

From an energetic viewpoint, there are two possibilities that can explain the reduction in ϵ whenever the transition point is crossed and a new well-organized, persistent jet emerges. Consider the eddy energy equation obtained by multiplying ψ_i with the eddy potential vorticity equation of layer i. Adding the two resulting equations, neglecting the biharmonic diffusion terms, and then integrating over the entire domain, yields

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where EKE, EAPE, C_BC, and C_BT denote the eddy kinetic energy, eddy available potential energy, baroclinic energy conversions, and barotropic energy conversions, respectively. The overbar represents the zonal mean. Because the left-hand side of (3) must vanish for a sufficiently long time average, there must be a balance between C_BC, C_BT, and the various energy dissipation terms. One could use this balance requirement to understand the above decline in ϵ if there was a mathematical relationship between ϵ and one of the energy conversion or dissipation terms. Such a precise mathematical expression exists only if linear potential vorticity damping is used with identical dissipation timescales in both layers. Obviously this is not the case in the present model. Nevertheless, we can obtain insight into this problem by noting in (3) that the thermal (Ekman) damping is linearly proportional to the EAPE (lower layer EKE). Assuming that the upper layer EKE is positively correlated both with the lower layer EKE and the EAPE, one can establish a qualitative relationship,

ϵτ_LC_BCC_BT(4)

where τ_L is some linear damping timescale. Although unrealistic, if as discussed above, potential vorticity damping is used, (4) is exact with τ_L/2 being the potential vorticity damping timescale. Therefore, within each set of experiments, where the values of the damping coefficients are held fixed, one can conclude that the decline of ϵ is due to either a decrease in (C_BC) or an increase in (−C_BT) or their combination.

One sees that C_BC essentially increases at a uniform rate from W_c = 8 to 28, including through the transition point. However, the barotropic conversion −C_BT increases nearly 30%–40% during the transition, reversing the trend that −C_BT decreases as W_c increases, away from the transition points. Although it is rather difficult to see in Fig. 3a, energy dissipation due to both the thermal and Ekman damping decreases slightly across the transition points, resembling the trend of ϵ. The similarity in the trend between ϵ and energy dissipation due to the thermal and Ekman damping is more ubiquitous in other sets of experiments, shown in Figs. 3b–d, adding confidence in the assumption that led to (4).

The energy budgets for the HB, LB, WD, and SD experiments are shown in Figs. 3b, 3c, 3d, and 3e, respectively. Having demonstrated the robustness of the behavior of ϵ during each of the transitions in Fig. 3a, the values of W_c in these additional experiments are varied only to the extent that transitions from one to two jets are captured except for the HB experiment.

The decline of ϵ associated with the transition is evident in HB and WD, but not the SD and LB experiments. In the HB and WD experiments, the decline in ϵ is due to the increase in −C_BT. In the SD case, although −C_BT slightly increases across the transition point, which lies between W_c = 8 and 9, ϵ does not decrease because the increase of C_BC is larger than that of −C_BT. This is consistent with the fact that the meridional shear of the jets is weaker for the SD and LB cases than the other cases, because −C_BT is proportional to the jet shear. Also, if drag is strong enough, the role of −C_BT in the eddy energy balance is expected to become smaller. For the HB experiment, above the transition point C_BC increases substantially, generating an even more complex behavior for ϵ. The reason for this large increase of C_BC is unclear. As stated above, for the LB experiment, no decline in ϵ is found. In this experiment, although the time-mean jet shows two distinct peaks for 18 ≤ W_c (Fig. 7b), a persistent two jet state was never realized; instead, one and two jet states alternate in an irregular manner. As will be discussed in section 4, we suspect that it is not the mere occurrence of two jets but the persistence of two jets that is associated with the decline in ϵ.

Lastly, we note that the transition points vary as the values of β and κ_M are changed. For example, the transition point lies between W_c = 10 and 11 for the control experiment, whereas it exists between W_c = 8 and 9 for the HB experiment. Comparing Figs. 3a–e, it is clear that the value of the transition point is smaller for the HB and SD experiments, relative to that of the control. Not surprisingly, since W_c measures the width of the baroclinically unstable region of the radiative equilibrium flow, just above the transition point, the jet scale in the HB and SD experiments is less than that for the control experiment (see Figs. 5 and 7). Such behavior is consistent with the findings of Panetta (1993), where it was shown that the jet scale in his QG turbulence simulation is well approximated by L_β = 2π(u_rms/β)^1/2; L_β is proportional to the rms eddy velocity u_rms and inversely proportional to β. In general, many studies find that the value of u_rms decreases as κ_M and β increase. Therefore, as β and κ_M increase, L_β decreases.

4. Role of the critical latitudes in multiple jets

Given the fact that the decline of ϵ associated with the jet transition is robust within a finite region of parameter space, in this section we explore the mechanism behind this behavior in more detail. Ultimately, this should lead to an improved understanding of the abrupt organization of multiple jets and their persistence.

Because the decline of ϵ is due to barotropic energy conversion from the waves to the zonal mean flow, one wonders whether the behavior of ϵ as well as the maintenance of the zonal jets can be understood in terms of wave absorption at critical latitudes, where the zonal-mean zonal wind velocity equals the waves’ phase velocity. Baroclinic life-cycle experiments (e.g., Feldstein and Held 1989) show that the eddy momentum flux divergence occurs near critical latitudes, thus decelerating the zonal mean wind in that region. Although it is yet to be shown, the systematic wave absorption associated with critical latitudes can explain the persistence of the multiple jets.

In order to investigate the possible role of the critical latitudes, we use phase speed spectra of the eddy fluxes, as described by Randel and Held (1991). For each latitude, cospectra of zonal mean eddy heat flux and zonal mean eddy momentum flux convergence are calculated as a function of zonal wavenumber and frequency. One can then calculate the sum of the contributions from all zonal wavenumbers and corresponding frequencies toward a specific zonal phase speed. The results of this procedure are presented as contour diagrams of wave covariance versus phase speed and latitude. For conciseness, such diagrams will be referred to as PL (phase velocity–latitude) diagrams throughout this paper.

5. Linear stability analysis

Although the relevance of linearly unstable modes to the atmosphere as well as to nonlinear models can be challenged in a number of different ways, in order to understand the interjet disturbances, a natural first step is to study linearly unstable modes. Figures 8a, 8b, 8c, and 8d, respectively, show the meridional structure of the streamfunction amplitude, heat flux, relative vorticity flux, and potential vorticity flux of the fastest growing normal mode of the time-mean flow for the control, W_c = 11 case. The growth rate and zonal wavenumber of this fastest growing normal mode are 4.6 × 10⁻² and 0.6, respectively. In both layers, the maximum streamfunction amplitude (Fig. 8a) occurs at y = 45, the interjet minimum. This mode grows baroclinically, and the heat flux (Fig. 8b) reaches its maximum at the interjet minimum. More importantly, the momentum flux divergence maximum also occurs at the interjet minimum, flanked by momentum flux convergence on either side at y = 43 and 47 (see Fig. 8c). Consistent with this heat and relative vorticity flux structure, the potential vorticity flux (Fig. 8d) in either layer peaks at the interjet minimum. We give the name “inter-jet mode” to normal modes with this distinct structure, in contrast to “conventional” normal modes that have their maximum heat flux and maximum momentum flux convergence at the jet maximum (see Figs. 9b and 9c).

This inter-jet mode is robust; for the W_c = 11 case of the control experiment, given any zonal wavenumber the fastest growing normal mode possesses essentially the same structure as that for k = 0.6 (see Fig. 8). The growth rate, phase speed, and structural characteristics of some selected unstable modes, including the fastest growing mode, are summarized in Table 2. For k ≤ 0.4 and k ≥ 1.0, there is no unstable mode.

The inter-jet mode is also the most unstable normal mode in each of the four selected runs from the HB, LB, WD, and SD experiments, described in section 4b. Furthermore, except for the LB case, the interjet mode is the fastest growing normal mode for any given zonal wavenumber. In the LB case, for k = 0.3, 0.4, and 0.5, none of the unstable modes take the form of the interjet mode. Recalling that two persistent jets are not found in the LB experiment, it is interesting to note that in this case the bulk of the wave energy in the nonlinear integration lies at k = 0.3, where an unstable interjet mode does not exist.

The structure of the the interjet mode compares very favorably with the interjet disturbances found in the nonlinear control calculation, for which the dominant zonal wavenumber at the interjet minimum is 0.5. As stated above, for k = 0.5, the fastest growing normal mode is the inter-jet mode with a phase speed of −0.432 (see Table 2). In Figs. 5a and 5b, along c = −0.432, the local heat flux and momentum flux divergence maxima are clearly seen to lie at y = 45. Furthermore, in Fig. 5b, the local momentum flux convergence maxima are present at y = 42 and 48, very close to those shown in Fig. 8c.

Unstable conventional normal modes, whose structure is similar to the disturbances growing on the jet maxima, are found only for k = 0.7 and k = 0.8. In both cases, these unstable modes are the second and third fastest growing modes. Figures 9a–d show the meridional structure of the streamfunction amplitude, heat flux, relative vorticity flux, and potential vorticity flux of the second fastest growing normal mode for k = 0.7. The largest amplitude of this mode is found at the southern jet maximum. The third fastest growing normal mode shows identical structure, except that its largest amplitude lies at the northern jet maximum (not shown). As noted earlier, the heat flux maximum occurs at the jet maximum; however, unlike the inter-jet mode, the momentum flux divergences are found away from the jet maximum, that is, at y ≈ 36 and 41. As a result, the upper-layer potential vorticity flux extrema also occur at the same latitudes, on either side of the jet (Fig. 9d). These latitudes where the greatest upper layer potential vorticity flux occur match with the critical latitudes of the nonlinear integration (see Fig. 5b).

6. Concluding remarks

This paper describes some aspects of quasigeostrophic turbulence in a parameter regime where spontaneously generated zonal jets exist. We focus on the transition from a single to a double jet state. When β is greater than a moderate value (≈0.2) and surface friction κ_M is not too strong (κ_M less than ≈0.3), the transition occurs abruptly as the width of the baroclinically unstable region of the initial flow is systematically increased. Across this transition point, a disorganized single meridionally meandering jet is replaced by two persistent jets that exhibit much less fluctuation. Close to the parameter value at which the transition occurs, the eddy energy ϵ is smaller for the double jet state than for the single jet. Such behavior goes against the general tendency of ϵ to increase away from the transition point. It is found that barotropic energy conversion from the eddies to the jets is responsible for the local minimum in ϵ in the vicinity of the transition point.

We believe that critical layer absorption plays a crucial role in the abrupt formation and maintenance of the multiple zonal jets, at least near the transition point. Just above the transition point where the flow organizes itself into two persistent jets, eddy momentum flux divergence maxima occur at all critical latitudes associated with both jets. Such a systematic zonal mean flow deceleration at the critical latitudes, caused by the eddy momentum flux divergence, results in a strong meridional shear of the zonal-mean zonal flow and hence increases the barotropic energy conversion from the eddy to the zonal mean flow. An exception is found in the LB experiment, where the transition from one to two jets no longer occurs abruptly. Instead, the transition is gradual and the temporal flow evolution is complex, being characterized by an alternation between one and two jet states. Spectral analysis indicates that the dominant zonal wavenumber has a negative phase speed. Hence there are no critical latitudes in the upper layer for this zonal wavenumber.

When two or more jets coexist, growing baroclinic waves are found to exist along the inter-jet minimum. These waves, referred to as inter-jet disturbances, possess the property that their momentum flux divergence occurs at the same latitude as their heat flux maximum. One might expect this result because a positive momentum flux divergence at an inter-jet minimum implies that the inter-jet disturbances are tilted along the shear of the zonal-mean zonal winds. However, this does not have to be the case. In fact, if the potential vorticity gradient of the basic flow changes sign in the vertical, which is indeed the case in our calculations, there is no obvious way to predict the direction of the vertically integrated momentum flux (Held 1975), although some general rules are found for a weakly nonlinear flow (e.g., Held and Andrews 1983). Because the momentum flux divergence decelerates the zonal jet at its local minimum, we suspect that the interjet disturbance plays an important role in maintaining the multiple jets by separating them from each other. In fact, away from the transition point, it appears that the inter-jet disturbance is the main player that decelerates the zonal flow between the two jets (see Fig. 6a). The structure of the heat and momentum fluxes of these inter-jet disturbances are remarkably well captured by the fastest growing normal mode. The inter-jet disturbance becomes more prominent as drag is reduced. Given its potential importance for maintaining multiple zonal jets, this inter-jet disturbance warrants further investigation. It is also worth noting that existence of this type of inter-jet disturbance is hinted at in a general circulation model (Williams 1988). In that study, when the rotation rate of the earth is increased by more than a factor of 2, more than two eddy-driven jets emerge. As Williams (1988) points out, the poleward eddy heat flux is continuous with latitude, whereas the eddy momentum flux shows clear convergence (divergence) at the jet maxima (interjet minimum). It would be of interest to construct the PL diagram with such calculations, in order to test the existence of inter-jet disturbances in more realistic models.

From the results of the analyses presented in this paper, it is not completely clear as to how the abrupt splitting of the jet occurs. Given an eddy meridional scale that is solely determined by the internal dynamics of the flow as the width of the baroclinically unstable region gradually increases, there must be some point at which two eddies can simultaneously fit in the meridional direction. When this condition is met, two jets can be spontaneously generated in association with these eddies. However, in forcing these two jets, we speculate that each eddy creates its own “cavity” because of the wave absorption at its critical latitudes, further isolating itself from the other eddy. This isolation is enhanced by the formation of the inter-jet disturbances. In this manner, two persistent jets may organize abruptly with only a small increase in the width of the baroclinically unstable region.

Although not presented in the text, the linear stability analysis of the initial flow cannot predict the number of jets in the statistically steady state. In particular, the structure and growth rate of normal modes growing on the initial flow are essentially the same on either side of the transition point. For example, in the control experiment with W_c = 10 and W_c = 11, the fastest growing normal mode is found at k = 7, with the structure of a conventional unstable normal mode.

To the extent that radiative forcing of the atmosphere and the action of the wind stress on the ocean surface are dynamically equivalent, the results of this paper imply the need for additional complexity in eddy flux and energy parameterizations, in particular for noneddy resolving ocean models.