## 1. Introduction

Zonal flows are banded, anisotropic, weakly fluctuating alternating jets that form spontaneously and persist indefinitely in an otherwise turbulent plasma or planetary fluid (Diamond et al. 2005; Vasavada and Showman 2005). The subject started with Rhines’ discovery that freely evolving barotropic *β*-plane turbulence transfers energy into zonal shear modes with zero frequency (Rhines 1975). Also in 1975, experiments by Whitehead showed that forcing, without the exertion of azimuthal torque, in a rapidly rotating basin produces prograde jets; in this context the curved upper surface provides an analog of the *β* effect. We follow Galperin et al. (2006) in referring to the development and persistence of these anisotropic planetary flows as “zonation.” Williams (1978) showed that zonation occurs in statistically steady forced-dissipative flows on the sphere and proposed this as an explanation of the banded structure of the planetary circulations of Jupiter and Saturn.

Figure 1 shows a typical example of fully developed, forced and dissipative zonation obtained by numerical solution of (3) below. The main features of the statistically steady flow, such as the sharp eastward jets, the broader westward return flows, and the sawtooth relative vorticity, are familiar from many earlier studies of statistically steady, stochastically forced, dissipative *β*-plane turbulence in doubly periodic geometry (Danilov and Gurarie 2004; Danilov and Gryanik 2004; Maltrud and Vallis 1991; Smith 2004; Vallis and Maltrud 1993) and on the sphere (Williams 1978; Nozawa and Yoden 1997; Huang and Robinson 1998; Scott and Polvani 2007).

**u**= (

*u*,

*υ*):

*f*=

*f*

_{0}+

*βy*is the

*β*-plane Coriolis frequency. The flow is energized by a solenoidal (incompressible) force generated by the function

*a*(

*x*,

*y*,

*t*). There is no loss of generality in taking the force to be solenoidal: any compressive component of the external force is balanced by the pressure gradient. Damping is provided by a combination of drag

*μ*and hyperviscosity

*ν*(with

_{n}*n*= 4 in numerical simulations, and

*n*= 1 in development of theory).

*ψ*(

*x*,

*y*,

*t*) with (

*u*,

*υ*) = (−

*ψ*,

_{y}*ψ*), and relative vorticity

_{x}*ζ*=

*ψ*+

_{xx}*ψ*. Eliminating the pressure from (1), one obtains the

_{yy}*β*-plane vorticity equation

*ξ*on the right-hand side of (11) is the curl of the solenoidal force in the momentum equation—that is,

We assume that the forcing, *a* in (1) and *ξ* in (3), is a rapidly decorrelating, isotropic, spatially homogeneous, random process. Thus energy and enstrophy are injected into a narrow band of wavenumbers centered on a “forced wavenumber” *k _{f}* (see appendix A for details of the implementation). This model of exogenous stochastic forcing, first proposed by Lilly (1969), is now a standard protocol used in many barotropic and shallow-water studies of forced-dissipative zonation. The physical interpretation of the forcing and the choice of its spatial structure vary somewhat in literature. Considering

*ξ*to be a representation of baroclinic eddies, Williams (1978) chose the forced wavenumbers to lie in a narrow rectangular band, with the zonal extent of the band equal to the baroclinic deformation radius. Scott and Polvani (2007) and Smith (2004) interpreted the rapidly decorrelating, narrowband, isotropic forcing as a model of small-scale three-dimensional convection. Another possibility is that

*ξ*is a representation of the bubble-cloud forcing used by Whitehead (1975) in the laboratory. Below, in the discussion surrounding (10) we give yet another interpretation of

*ξ*.

We have found no studies that establish any particular forcing protocol as being a reasonable physical representation of three-dimensional small-scale eddies acting on a barotropic flow. However, despite the two strong modeling approximations, namely quasi-geostrophy and the choice of the forcing, some features of Jovian jets, such as the jet width, are approximately captured by simplified models (Smith 2004; Vasavada and Showman 2005). A different model forcing in Showman (2007) uses nonsolenoidal physical space mass forcing to represent moist convection in a shallow water system. Despite the different choice of forcing, Showman’s results on planetary zonal jets are broadly consistent with those obtained by Smith (2004). In light of this fact, and since the barotropic quasigeostrophic (QG) system cannot represent mass forcing, we do not address these issues further.

A common theme in all the studies mentioned above is a separation of scales between the forcing length scale

A striking feature of *β*-plane zonation is that the translational symmetry, *y* → *y* + *a*, of the equation of motion (3) is spontaneously broken: the locations of the eastward maxima in Fig. 1 are an accident of the initial conditions and of the random number generator used to create *ξ*. But after the jets form, they remain in the same position, apparently forever. Once these robust quasi-steady jets are in place, their dynamics can be discussed in mechanistic terms using concepts such as potential vorticity (PV) mixing, the resilience of transport barriers at the velocity maxima, radiation stress, and shear straining of turbulent eddies (Rhines and Young 1982; Dritschel and McIntyre 2010). But the primary question addressed here is why the jets form in the first place, given that *ξ* does not select particular locations. Following earlier investigations of this phenomenon (Farrell and Ioannou 2007; Manfroi and Young 1999), we show that zonation can be understood as symmetry-breaking instability of an isotropic, spatially homogeneous, and jetless *β*-plane flow.

In section 2 we introduce the eddy-mean decomposition and discuss a statistical method, previously used by Farrell and Ioannou (1993b, 2003, 2007), Marston et al. (2008), and Tobias et al. (2011), which is the basis of our linear stability analysis of zonostrophic instability. This method amounts to forming quadratic averages of the equations of motion and then discarding third-order cumulants. Farrell and Ioannou (2003, 2007) refer to this method as stochastic structural stability theory (SSST), while Marston et al. (2008) call it the second-order cumulant expansion, or CE2. SSST and CE2 are completely equivalent, and only one name is required. We have therefore adopted the more descriptive CE2 terminology of Marston et al. (2008).

In section 3 we present a physical space reformulation of CE2, which has analytic advantages over earlier numerically oriented formulations. Within the context of CE2, section 4 provides a complete analytic description of zonostrophic instability obtained by linearizing around an exact isotropic and homogeneous solution with no jets. As in Farrell and Ioannou (2007), zonation is understood as a linear instability of CE2: part of the linearly unstable eigenmode is a zonal flow. This linear stability problem is characterized by two control parameters, a nondimensional drag parameter *μ*_{*} and a nondimensional planetary parameter *β*_{*}, and we determine the CE2 zonostrophic stability boundary in the (*β*_{*}, *μ*_{*})-parameter plane. An important property of CE2 zonostrophic instability is that the most unstable wavenumber, which determines the meridional scale of the exponentially growing jets, is well away from zero. Because the instability unfolds around a nonzero wavenumber, CE2 zonostrophic instability is not properly a negative-viscosity instability. This point is reinforced in section 5 by showing that the CE2 eddy viscosity is identically zero. Section 6 is a comparison between the analytic results and direct numerical simulations of the nonlinear system. Section 7 is the discussion and conclusions. The more technical aspects of the paper are in five appendices.

## 2. The eddy-mean decomposition and quasilinear dynamics

*n*= 1 for the viscosity.

### a. Quasilinear dynamics

Quasilinear (QL) zonal jets. (left) A snapshot of the zonally averaged velocity *U*(*y*, *t*) obtained by integrating the QL system (6), (7), and (9). (right) A snapshot of the QL vorticity *ζ*, with overlaid zonally averaged vorticity −*U _{y}* (solid white curve). The parameters for this run are the same as the nonlinear solution in Fig. 1 (i.e.,

*μ*

_{*}= 0.0182,

*β*

_{*}= 1, and

*k*= 32). The snapshot is at 2

_{f}L*μt*= 40 after spinup from rest.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Quasilinear (QL) zonal jets. (left) A snapshot of the zonally averaged velocity *U*(*y*, *t*) obtained by integrating the QL system (6), (7), and (9). (right) A snapshot of the QL vorticity *ζ*, with overlaid zonally averaged vorticity −*U _{y}* (solid white curve). The parameters for this run are the same as the nonlinear solution in Fig. 1 (i.e.,

*μ*

_{*}= 0.0182,

*β*

_{*}= 1, and

*k*= 32). The snapshot is at 2

_{f}L*μt*= 40 after spinup from rest.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Quasilinear (QL) zonal jets. (left) A snapshot of the zonally averaged velocity *U*(*y*, *t*) obtained by integrating the QL system (6), (7), and (9). (right) A snapshot of the QL vorticity *ζ*, with overlaid zonally averaged vorticity −*U _{y}* (solid white curve). The parameters for this run are the same as the nonlinear solution in Fig. 1 (i.e.,

*μ*

_{*}= 0.0182,

*β*

_{*}= 1, and

*k*= 32). The snapshot is at 2

_{f}L*μt*= 40 after spinup from rest.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparing the left panels in Figs. 1 and 2, one sees that the QL jets are faster and wider than NL jets, and the jet profiles are different: QL jets are distinctly more east–west symmetric than NL jets. Nonetheless, we show in section 6 that the QL jets in Fig. 2 do have a small east–west QL asymmetry, and at other points in the (*β*_{*}, *μ*_{*})-parameter space, QL jets are strongly east–west asymmetric.

Because the QL jets are faster, the QL system is more zonostrophically unstable than the NL system. In Figs. 1 and 2, quasi-steady jets evolve spontaneously from an initially jetless state, as shown in the Hovmöller diagram of the zonal mean flow in Fig. 3. Comparing Figs. 3a and b shows that the QL system has significantly longer adjustment times than the NL system.

(a) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the full NL system in (3). (b) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the QL system. (c) A comparison of the zonal mean energy fraction, zmf(*t*) defined in (78), for QL and NL runs. The time-averaged fractions are 〈zmf〉_{NL} = 0.3 and 〈zmf〉_{QL} = 0.51. This figure shows the time evolution of the runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the full NL system in (3). (b) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the QL system. (c) A comparison of the zonal mean energy fraction, zmf(*t*) defined in (78), for QL and NL runs. The time-averaged fractions are 〈zmf〉_{NL} = 0.3 and 〈zmf〉_{QL} = 0.51. This figure shows the time evolution of the runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the full NL system in (3). (b) Hovmöller diagram of the zonal mean velocity *U*(*y*, *t*) obtained by solution of the QL system. (c) A comparison of the zonal mean energy fraction, zmf(*t*) defined in (78), for QL and NL runs. The time-averaged fractions are 〈zmf〉_{NL} = 0.3 and 〈zmf〉_{QL} = 0.51. This figure shows the time evolution of the runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

O’Gorman and Schneider (2007) made the QL approximation (9) in an atmospheric general circulation model and showed by comparison with the full nonlinear version of the model that several important features of the flow are unaffected by complete removal of the eddy–eddy nonlinearity as in (9). Comparing Figs. 1 and 2 we reach a similar conclusion for the more idealized model studied here. This preliminary conclusion is supported by a detailed comparison between NL and QL solutions in section 6.

There are several ways of motivating QL dynamics. The QL system conserves both energy and enstrophy and has the same zonal mean equation and symmetries as the NL system. Thus, arguments based on quadratic integral invariants apply equally to the QL and the NL system (Salmon 1998). Nonetheless, because EENL is discarded, the QL system cannot exhibit a true Batchelor–Kraichnan inverse energy cascade: in the QL model all nonlinear interactions require participation of the zonal mean flow. Because *U*(*y*, *t*) has a larger length scale than the eddy field, all these nonlinear QL interactions are spectrally nonlocal. Fig. 2 shows that the spectrally local Batchelor–Kraichnan inverse cascade is not necessary for zonation.

PV is not materially conserved by the QL system, and consequently nonquadratic functions of PV are not conserved by the nonlinear terms remaining in QL. Thus, Fig. 2 also shows that strict material conservation of PV is not necessary for zonation.

Thus, at the most basic level, the QL system is instructive as an indication of the physically essential processes necessary for zonation.

### b. Stochastic closure versus cumulant expansion

*ξ*

_{EENL}(

**x**,

*t*) and the dissipation

*μ*

_{EENL}

*ζ*′ should be chosen to match the evolution of the NL system. The terms in (10) then augment the exogenous forcing and dissipation on the right-hand side of (7).

However, there is probably no reliable a priori method of determining the right-hand side of (10). Heeding the principle to first do no harm, we prefer the QL alternative (9). This has the advantage that one can then make a specific comparison between QL and NL solutions (e.g., as in Figs. 1 and 2) and assess the role of EENL.

Our point of view, which follows Marston et al. (2008) and Tobias et al. (2011), is to regard the QL system as an approximation to the NL system. In fact, (9) in tandem with the method of Farrell and Ioannou, is precisely the second-order cumulant expansion CE2 of Marston et al. (2008). It is from this perspective that in section 3 we develop a physical-space formulation of CE2, which is suitable for analytic solution.

## 3. Dynamics of correlations: CE2

*ζ*′ and streamfunction

*ψ*′. This correlation equation [(23) below] is coupled to the evolution of the zonal mean flow via the Reynolds stresses, and the Reynolds stresses can be obtained by evaluating derivatives of the correlation function at zero separation. Thus one obtains the zonal mean evolution equation in (34) below.

### a. Correlation functions: Kinematics

*a*(

**x**,

*t*) in (1) and

*ξ*(

**x**,

*t*) in (11), has two-point, two-time correlation functions of the form

*A*and Ξ only on the difference

*A*and Ξ are related by

*n*is

Notice that that we have changed notation: undecorated *x* in (15) is the zonal difference coordinate. We also use the shorthand *n* = 1 or 2 to explicitly indicate whether we refer to the eddy vorticity equation at the point **x**_{1} = (*x*_{1}, *y*_{1}) or at the point **x**_{2} = (*x*_{2}, *y*_{2}). We forbear from doing so.

*x*,

*y*

_{1},

*y*

_{2},

*t*), one can obtain the velocity correlation tensor as

**x**

_{1}and the other as

**x**

_{2}is arbitrary, all correlation function have an important “exchange” symmetry

*A*,

### b. Correlation functions: Dynamics

*n*is

*x*

_{1}and

*x*

_{2}only through the combination

*x*=

*x*

_{1}−

*x*

_{2}. Because of this zonal homogeneity

*δ*(

*t*

_{1}−

*t*

_{2}) in (14). Considerations summarized in appendix B (amounting to a simple proof of Ito’s formula) show that

### c. Collective coordinates

*y*

_{1}and

*y*

_{2}, there are advantages in using the “collective coordinates”

### d. The zonal mean flow equation

One advantage of collective coordinates is that mean square quantities, such as the enstrophy, are obtained by evaluating correlation functions at zero separation [i.e., by setting (*x*, *y*) = 0]. For example, if one possesses

## 4. Zonostrophic instability of a spatially homogeneous and isotropic base-state flow

### a. The spatially homogeneous basic state

*r*is the two-point separation defined in (27). Because Ξ does not depend on

*U*= 0. With these simplifications the correlation equation (31) collapses to

*H*emphasizes that

_{H}(

*x*,

*y*) can be obtained from

*β*: an isotropic and spatially homogenous forcing drives an isotropic and spatially homogeneous flow, despite the anisotropy of Rossby wave propagation.

*h*→ 0. To see this, we recall (16), which in this homogeneous and isotropic case implies that Ξ = ∇

^{4}

*A*, and therefore

*a*(

**x**,

*t*) is stationary, the spectra

*h*→ 0 (provided that

*μ*≠ 0).

^{4}

*A*, and the assumption that the correlation function

*A*(

*r*) decays faster than

*r*

^{−1}as

*r*→ ∞. The constraints above are satisfied by the standard forcing protocol described in appendix A, which has zero spectral density around

*h*= 0.

### b. The dispersion relation of inviscid and isotropic flow

_{H},

*m*is the meridional wavenumber of the disturbances and

*s*is the growth rate, with growing perturbations corresponding to ℜ(

*s*) > 0. Retaining terms linear in the perturbation variables

*Q*(

*χ*,

*n*) is defined by the angular integral

*A*=

*A*(

*r*) and

*ν*= 0; a more general expression of the dispersion relation is in appendix C.

Dr. George Carnevale has shown that the dispersion relation in (45) and (46) is also obtained from (5.13) in Carnevale and Martin (1982). The field-theoretic approach of Carnevale and Martin (1982) is different from the approximation used to obtain the CE2 system in (31) and (34); for instance, CE2 contains terms such as *U* = 0, these terms are neglected. Therefore, the linearized version of CE2 in this section is equivalent to the weak-turbulence limit (5.13) in Carnevale and Martin (1982). This consistency provides confidence in (45) and (46).

### c. Ring forcing

*h*=

*k*and has thickness 2

_{f}*δk*≪

*k*. This is the “narrow-band forcing” described in appendix A. We idealize this choice further by considering “ring” forcing corresponding to the limit

_{f}*δk*→ 0. In other words, we consider a random flow, driven isotropically by injecting energy on the circle

*h*=

*k*in wavenumber space. This corresponds to

_{f}*J*

_{0}is the Bessel function of order zero. Notice that

*ν*= 0—as we assume in (45)—the spatially homogeneous base-state solution in (41) is

*ε*above, with dimensions of watts per kilogram, is the rate of working of the force that sustains the base-state (48) flow against dissipation.

*δ*(

*h*−

*k*) in (47), the

_{f}*h*integral in (45) is trivial. Before proceeding, however, it is convenient to write the various parameters in the nondimensional form using the length scale

*Q*defined in (46). We now lighten the notation by dropping the asterisk on nondimensional variables

*m*and

*s*. We have obtained the growth rate by solving (51) numerically for

*s*=

*s*+

_{r}*is*. This numerical solution indicates that modes with

_{i}*s*> 0 are found only if 0 <

_{r}*m*

^{2}< 1, and these unstable modes have

*s*= 0. We have been unable to obtain a satisfactory nonnumerical proof of these two important properties of zonostrophic instability.

_{i}*s*(

*m*) plotted as a function of

*m*for various values of

*β*

_{*}, with

*μ*

_{*}= 0.15 in all cases. If

*s*> 0 for some values of

*m*(e.g.,

*β*

_{*}= 0.15 and 1 in Fig. 4), then the homogeneous flow is unstable and zonal jets will grow from very small initial amplitude. Also shown in Fig. 4 are two marginally stable situations

*β*

_{*}= 0.0634 and

*β*

_{*}= 2.571. These are defined by the condition that the most unstable disturbance has

*s*= 0:

*β*

_{*}is either too large or too small (e.g., in Fig. 4 the flow is stable if

*β*

_{*}> 2.571 or if

*β*

_{*}< 0.0634).

The growth rate *s* as a function of *m* for *μ*_{*} = 0.15 and five values of *β*_{*} indicated on the curves. The variables in this figure are nondimensionalized according to (49) and (50). These modes have *s _{i}* = 0 (i.e.,

*s*is real). The curves

*β*

_{*}= 2.571 and 0.0634 correspond to the marginally stable situation defined by (52).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The growth rate *s* as a function of *m* for *μ*_{*} = 0.15 and five values of *β*_{*} indicated on the curves. The variables in this figure are nondimensionalized according to (49) and (50). These modes have *s _{i}* = 0 (i.e.,

*s*is real). The curves

*β*

_{*}= 2.571 and 0.0634 correspond to the marginally stable situation defined by (52).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The growth rate *s* as a function of *m* for *μ*_{*} = 0.15 and five values of *β*_{*} indicated on the curves. The variables in this figure are nondimensionalized according to (49) and (50). These modes have *s _{i}* = 0 (i.e.,

*s*is real). The curves

*β*

_{*}= 2.571 and 0.0634 correspond to the marginally stable situation defined by (52).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

*β*

_{*},

*μ*

_{*}) parameter plane. This curve,

*β*

_{*},

*μ*

_{*}) parameter space (i.e., the largest value of drag

*μ*

_{*}at which the homogeneous solution loses stability). The lower panel of Fig. 5 shows the wavenumber

*m*(

^{c}*β*

_{*}) of the incipient instability [i.e., the wavenumber determined by simultaneously satisfying the two equations in (53)].

(a) The critical curve *β*_{*} ≫ 1 (dash-dotted) and *β*_{*} ≪ 1 (dashed) are shown in both panels.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a) The critical curve *β*_{*} ≫ 1 (dash-dotted) and *β*_{*} ≪ 1 (dashed) are shown in both panels.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a) The critical curve *β*_{*} ≫ 1 (dash-dotted) and *β*_{*} ≪ 1 (dashed) are shown in both panels.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

### d. Approximations to the neutral curve with large and small β_{*}

*β*

_{*}≪ 1 and

*β*

_{*}≫ 1. These results are obtained via asymptotic analysis of the integral

*Q*(

*χ*,

*n*) in (46), and simplification of the dispersion relation [(51); see appendix E]. If

*β*

_{*}≪ 1 then the critical curve is

*β*

_{*}≫ 1, the approximation to the critical curve is

The lower panel of Fig. 5 shows that linear zonostrophic instability is spectrally nonlocal only in the limit *β*_{*} → ∞: in that case the most unstable wavenumber is much less than the forced wavenumber *k _{f}*, implying a scale separation between the scales at which energy is injected and the scale at which jets initially form. In the other limit

*β*

_{*}→ 0 the linearly unstable wavenumber is close to

*k*.

_{f}### e. The small wavenumber structure of the growth rate

*m*provides insight into the nature of zonostrophic instability. Looking at Fig. 4, we anticipate that

*η*

_{2}> 0 might explain the increase in

*s*that results in the instability with

*s*> 0. This would be a “negative-viscosity instability,” which is the interpretation offered by Farrell and Ioannou (2007), Bakas and Ioannou (2011, manuscript submitted to

*J. Atmos. Sci.,*hereafter BakIo), and Bakas and Ioannou (2011).

*m*= 0 is

*η*

_{2}in (58), corresponding to viscosity, is identically zero. Instead, the instability is associated with a destabilizing hyperviscous term, namely the Reynolds stresses are related to the zonal mean flow by

*m*growth rate in (59). We analyze this curious situation further in section 5 and show that

*η*

_{2}= 0 follows from the assumed isotropy of the forcing [i.e.,

*η*

_{2}= 0 is not a special property of the particular model in (48)]. The conclusion is that zonostrophic instability requires antifrictional momentum fluxes, and in the small-

*m*limit this antifriction is hyperviscous.

In recent work BakIo and Bakas and Ioannou (2011) reach a different conclusion, namely that the antifrictional effect resulting in nonzero Reynolds stress is equivalent to nonzero and positive *η*_{2}, and that the hyperviscous coefficient *η*_{4} is negative and therefore stabilizing. We believe that these differences may result from a different choice of forcing Ξ. Bakas and Ioannou (2011) and BakIo use an anisotropic forcing, while our conclusion above is specifically for isotropic forcing. The importance of isotropy to our conclusion is underscored in the section 5.

## 5. Isotropy and zero eddy viscosity

In the discussion surrounding (59) we observed that the term in the zonostrophic dispersion relation corresponding to the eddy viscosity is zero. This result emerges from the analysis of a complicated dispersion relation and surely deserves a more fundamental explanation, or at least another explanation. Thus in this section we more directly obtain the eddy viscosity of an isotropically forced QL *β*-plane shear flow and show that the result is identically zero.

*U*=

_{n}*γy*, and in this case the CE2 correlation equation (31) collapses to

_{n}*ν*is defined by the relation

_{e}_{xy}at zero separation [e.g., as in (33)]. The eddy viscosity then follows from definition (62).

We expect that *ν*_{e} defined above is equal to the coefficient *η*_{2} in (58). In the *m* → 0 limit, the modal solution in (44) varies on the length scale *m*^{−1}, which is much greater than the length scale of the forcing, namely ^{1} (except at the “shearless” points, where *U _{y}* = 0). By calculating the Reynolds stress in this situation one can anticipate the low-wavenumber structure of the dispersion relation. This reasoning is identical to methods in kinetic theory by which the molecular shear viscosity is calculated.

### a. A solution of the correlation equation

*β*effect is removed from the problem. Equation (63) can be solved straightforwardly as an ordinary differential equation in

*x*. However, to make contact with a large literature on sheared disturbances, it is instructive to consider the initial-value problem

*F*, the vorticity correlation function is written as the time integral of a sheared disturbance:

### b. The Reynolds stresses

^{4}

*G*=

*δ*(

*x*)

*δ*(

*y*), or

*G*(

*x*,

*y*) in hand, we have

_{xy}at zero separation, or

*t*integral is last, and in the inner

*x*and

*y*integrals “unshear” the correlation function with the coordinate change:

We remark that the constraints in (42) and (43) are required so that correlation function Ψ on the left of (69) decays as *r* → ∞, despite the *r* → ∞ divergence of the Green’s function *G*(*r*) in (68). In the convolution integral on the right-hand side of (69), the large *r* divergence of *G* is shielded by zero integrals of the vorticity correlation function

There are two important caveats associated with the conclusion that *ν _{e}* = 0: the stochastic forcing is isotropic and dissipation is provided only by Ekman drag. Relaxing either or both of these assumptions might result in nonzero

*ν*.

_{e}### c. The kinetic energy density

*β*-plane Couette flow problem considered here is obtained by first rewriting (63) as

^{2}

*ε*/(2

*μ*), independent of both

*β*and

*γ*.

### d. Discussion

To a certain extent the result *ν _{e}* = 0 is anticipated in the literature on sheared disturbances. Shepherd (1985) showed that an isotropic initial distribution of Rossby waves maintains a constant energy density, despite shearing by a Couette flow; see also Farrell and Ioannou (1993a) and Holloway (2010). The solution in (67), with the isotropic initial condition in (65), is essentially a time integral of Shepherd’s solution of the sheared-disturbance problem with an isotropic initial condition.

Via direct numerical simulation (but with *β* = 0), Cummins and Holloway (2010) have recently shown that the eddy–eddy nonlinearity is essential in producing nonzero Reynolds stresses from Couette-sheared eddies. Cummins and Holloway (2010) identify the essential role of EENL as restoration of isotropy at high wavenumbers. Moreover, as a result of nonlinearly restored isotropy, *ν _{e}* is robustly positive and thus cannot serve as an explanation of zonostrophic instability. Whatever the sign of

*ν*, an unfortunate consequence of (9) is that restoration of isotropy at small scales is absent in QL dynamics and not represented in the ensemble-averaged dynamics CE2.

_{e}## 6. Zonation in QL and NL solutions

We now turn to numerical solutions for a comparison of the full nonlinear system, the quasilinear system, and the predictions of CE2. In these calculations the resolution is 512 × 512, and we use the ETDRK4 time-stepping scheme (Cox and Matthews 2005). In addition to the control parameters *β*_{*} and *μ*_{*} defined in (49), there is a third control parameter, which is the size of the domain relative to the forced wavenumber *k _{f}*: in our computations the domain is a doubly periodic square 2

*πL*× 2

*πL*, with

*k*= 32. Thus there is scale separation between the forcing and the domain.

_{f}L### a. The onset of zonation in NL and QL solutions

*t*) is a gross measure of the strength of the zonal mean flow. The time average, denoted by 〈zmf〉, is computed by averaging over an interval

*t*

_{1}<

*t*<

*t*

_{1}+ 10/

*μ*, where typically 2

*μt*

_{1}> 40. This long spinup ensures that statistical equilibrium has been achieved and is consistent with the equilibration time suggested by Galperin et al. (2006).

The index 〈zmf〉 is used to classify the flow. Figure 6 summarizes a suite of QL and NL calculations in which the drag parameter is varied at fixed *β*_{*}. The onset of zonation is indicated by the increase in 〈zmf〉. The dotted lines marked

The time-averaged zonal mean energy fraction 〈zmf〉 as a function of *μ*_{*}, with *β*_{*} fixed as indicated in the bottom-right corner of each panel. QL simulations are indicated by a degree sign and NL solutions by an asterisk.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The time-averaged zonal mean energy fraction 〈zmf〉 as a function of *μ*_{*}, with *β*_{*} fixed as indicated in the bottom-right corner of each panel. QL simulations are indicated by a degree sign and NL solutions by an asterisk.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The time-averaged zonal mean energy fraction 〈zmf〉 as a function of *μ*_{*}, with *β*_{*} fixed as indicated in the bottom-right corner of each panel. QL simulations are indicated by a degree sign and NL solutions by an asterisk.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The onset of zonostrophic instability requires significantly smaller values of *μ*_{*} in the NL case than in the QL case: in Fig. 6 the ratio

### b. Zonostrophically stable NL solutions

*μζ*≈

*ξ*and the vorticity field closely resembles a snapshot of the forcing

*ξ*.

Snapshots of the vorticity *ζ*(*x*, *y*, *t*) with overlaid zonally averaged vorticity −*U _{y}*(

*y*,

*t*) (solid white curve) with (a)

*μ*

_{*}= 0.309 and (b)

*μ*

_{*}= 0.0545. Both snapshots are at nondimensional time 2

*μt*= 25, after spinup from rest, and

*β*

_{*}= 1.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Snapshots of the vorticity *ζ*(*x*, *y*, *t*) with overlaid zonally averaged vorticity −*U _{y}*(

*y*,

*t*) (solid white curve) with (a)

*μ*

_{*}= 0.309 and (b)

*μ*

_{*}= 0.0545. Both snapshots are at nondimensional time 2

*μt*= 25, after spinup from rest, and

*β*

_{*}= 1.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Snapshots of the vorticity *ζ*(*x*, *y*, *t*) with overlaid zonally averaged vorticity −*U _{y}*(

*y*,

*t*) (solid white curve) with (a)

*μ*

_{*}= 0.309 and (b)

*μ*

_{*}= 0.0545. Both snapshots are at nondimensional time 2

*μt*= 25, after spinup from rest, and

*β*

_{*}= 1.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Figure 8 compares energy spectra of statistically steady QL and NL solutions. With strong drag (i.e., *μ*_{*} = 0.309) only the directly forced wavenumbers are significantly excited. As *μ*_{*} is reduced there is transfer of energy to small wavenumbers. In the NL case the transfer of energy to wavenumbers smaller than *k _{f}* is the due to the inverse energy cascade. In the QL case the excitation of small wavenumbers is due only to shearing by the zonal mean flow. Comparing QL and NL solutions at the same value of

*μ*

_{*}, one sees from Figs. 8b and 8d that there is significantly more low-wavenumber eddy energy in the NL cases. Yet the zonal mean energy is always stronger in the QL case. There is no clear association between the inverse energy cascade and zonation.

(a),(c) The zonal spectrum *E _{Z}*(

*k*/

_{y}*k*) for QL and NL solutions with

_{f}*β*

_{*}= 1. (b),(d) The residual spectrum

*E*(

_{R}*k*/

*k*), defined as the angularly averaged spectrum after removal of the “zonal modes” with

_{f}*k*= 0. The largest peak in

_{x}*E*(

_{Z}*k*/

_{y}*k*) defines the wavenumber

_{f}*m*, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with

_{Z}*μ*

_{*}= 0.0545 in (a)].

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a),(c) The zonal spectrum *E _{Z}*(

*k*/

_{y}*k*) for QL and NL solutions with

_{f}*β*

_{*}= 1. (b),(d) The residual spectrum

*E*(

_{R}*k*/

*k*), defined as the angularly averaged spectrum after removal of the “zonal modes” with

_{f}*k*= 0. The largest peak in

_{x}*E*(

_{Z}*k*/

_{y}*k*) defines the wavenumber

_{f}*m*, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with

_{Z}*μ*

_{*}= 0.0545 in (a)].

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

(a),(c) The zonal spectrum *E _{Z}*(

*k*/

_{y}*k*) for QL and NL solutions with

_{f}*β*

_{*}= 1. (b),(d) The residual spectrum

*E*(

_{R}*k*/

*k*), defined as the angularly averaged spectrum after removal of the “zonal modes” with

_{f}*k*= 0. The largest peak in

_{x}*E*(

_{Z}*k*/

_{y}*k*) defines the wavenumber

_{f}*m*, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with

_{Z}*μ*

_{*}= 0.0545 in (a)].

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

The NL solution shown in right panel of Fig. 7 with *μ*_{*} = 0.0545 has an eddy energy spectrum in Fig. 8b exhibiting the beginning of *β*-plane flow, without jets. To activate zonostrophic instability the drag must be reduced (e.g., to *μ*_{*} = 0.0182 in Figs. 1 and 8).

### c. The jet scale

If zonation occurs, as evinced by significantly nonzero values of 〈zmf〉, then by counting the number of distinct jets one can reliably estimate^{3} a jet wavenumber *m _{J}*. For example, in Fig. 1 there are seven jets and therefore

*E*(

_{Z}*k*/

_{y}*k*) has a strong peak: an example is the

_{f}*μ*

_{*}= 0.0545 solution in Fig. 7b: the corresponding zonal energy spectrum in Fig. 8a has a distinct peak even though there are no zonal jets. In cases like this, we report a wavenumber

*m*that is the peak of the zonal spectrum

_{Z}*E*(

_{Z}*k*/

_{y}*k*). In cases where there are strong jets we invariably find that

_{f}*m*≈

_{Z}*m*. It is interesting to compare

_{J}*m*and

_{J}*m*with a Rhines wavenumber defined as

_{Z}*υ*′. We found, however, that

*V*

_{RMS}gave the best estimate of the NL jet spacing at small values of

*μ*

_{*}. An advantage of

*V*

_{RMS}is that the energy power integral

^{4}can be used to express

*V*

_{RMS}in terms of external parameters as

Figure 9 compares the zonal wavenumber obtained from QL and NL solutions with the Rhines wavenumber on the right-hand side of (83), and with the most unstable wavenumber obtained from the linear stability analysis of section 4. In Fig. 9 we show only the *β*_{*} = 1 and *β*_{*} = 0.5 solutions: solutions with other values of *β*_{*} exhibit a broadly similar dependence of *m _{Z}* on

*μ*

_{*}.

A summary of zonal wavenumbers (jet scales) for solutions with (a) *β*_{*} = 0.5 and (b) *β*_{*} = 1. The dot-dashed curve is the Rhines wavenumber defined in (83). The solid curve labeled QLS is most unstable wavenumber calculated from the dispersion relation (51). The NL solutions are indicated by asterisks and the QL solutions by degree signs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

A summary of zonal wavenumbers (jet scales) for solutions with (a) *β*_{*} = 0.5 and (b) *β*_{*} = 1. The dot-dashed curve is the Rhines wavenumber defined in (83). The solid curve labeled QLS is most unstable wavenumber calculated from the dispersion relation (51). The NL solutions are indicated by asterisks and the QL solutions by degree signs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

A summary of zonal wavenumbers (jet scales) for solutions with (a) *β*_{*} = 0.5 and (b) *β*_{*} = 1. The dot-dashed curve is the Rhines wavenumber defined in (83). The solid curve labeled QLS is most unstable wavenumber calculated from the dispersion relation (51). The NL solutions are indicated by asterisks and the QL solutions by degree signs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

At large values *μ*_{*} only the directly forced modes are excited, and consequently *m _{Z}* ≈

*k*in both the QL and NL cases. At the critical value

_{f}*μ*

_{*}= 0.2 and 0.165 in Fig. 9a) the QL

*m*agrees with the analytic result from Fig. 5. In this regime the NL solutions start to develop an inverse cascade (but without exciting zonal jets) and the NL

_{Z}*m*begins to decrease.

_{Z}There is an interesting transition at *μ*_{*} is reduced the QL and NL wavenumbers are locked together. At *μ*_{*} in Fig. 9, which corresponds to the runs in Figs. 1 and 2, the QL and NL wavenumbers are almost equal and are estimated roughly by *m*_{Rh}.

In Fig. 9 the analytic result QLS agrees with the observed QL jet scale only when *μ*_{*} is not too far from the linear stability boundary *μ*_{*} significantly less than

Hovmöller diagrams for the (a) NL and (b) QL runs with *β*_{*} = 1.0 and *μ*_{*} = 0.0545. The NL run corresponds to the vorticity snapshot shown in Fig. 7b and shows zonal “streaks.” In (b) the QL jets initially appear at a wavenumber predicted by linearization of CE2. Then successive mergers result in an increase in jet spacing.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Hovmöller diagrams for the (a) NL and (b) QL runs with *β*_{*} = 1.0 and *μ*_{*} = 0.0545. The NL run corresponds to the vorticity snapshot shown in Fig. 7b and shows zonal “streaks.” In (b) the QL jets initially appear at a wavenumber predicted by linearization of CE2. Then successive mergers result in an increase in jet spacing.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Hovmöller diagrams for the (a) NL and (b) QL runs with *β*_{*} = 1.0 and *μ*_{*} = 0.0545. The NL run corresponds to the vorticity snapshot shown in Fig. 7b and shows zonal “streaks.” In (b) the QL jets initially appear at a wavenumber predicted by linearization of CE2. Then successive mergers result in an increase in jet spacing.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Figure 10a shows the Hovmöller diagram of the jetless NL solution from Fig. 7b. There is no zonation and *U*(*y*, *t*) shows “streaks” rather than jets. These streaks are not strong relative to the turbulent eddy field (i.e., 〈zmf〉 ≈ 0). The corresponding zonal energy spectrum in Fig. 8a exhibits a strong peak, which is a signature of these transient zonal steaks.

Figure 10b shows the QL case in which jets initially appear with a relatively small meridional spacing predicted by linear theory, followed by a sequence of mergers so that the mature flow has *m _{Z}* much less than the most linearly unstable wavenumber. The QL jet-merging phenomenology, which is effectively a one-dimensional inverse cascade, is very similar to the “Cahn–Hilliard” solutions obtained by Manfroi and Young (1999) from a model of deterministically forced zonation.

### d. The small drag regime

The flows in Figs. 1 and 2 have relatively light damping and both flows have organized jets containing a substantial fraction of the total kinetic energy. Figure 11 shows the time-averaged zonal mean flow 〈*U*〉 and the corresponding PV gradient *β*_{*} − 〈*U _{yy}*〉. In Fig. 2 the QL jets are almost symmetrical in the zonal direction, in contrast to the NL jets.

^{5}But the QL jets are not perfectly symmetric: the PV gradient in Fig. 11b reveals the QL east–west asymmetry. The NL PV gradient is positive for all

*y*and thus the NL jets are stable according to the Rayleigh–Kuo criterion. The QL PV gradient in Fig. 11b reverses sign on the flanks of the eastward jet, and also at the centers of the westward jets. Nonetheless the QL zonal mean flow shows no indication of barotropic instability [i.e., the deep spikes with

*β*

_{*}− 〈

*U*〉 < 0 are permanent features of the QL zonal mean flow even after time averaging].

_{yy}Comparison of zonal mean velocity profiles of the *β*_{*} = 1 NL and QL runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of zonal mean velocity profiles of the *β*_{*} = 1 NL and QL runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of zonal mean velocity profiles of the *β*_{*} = 1 NL and QL runs in Figs. 1 and 2.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

^{6}two nondimensional parameters

*β*

_{*}and

*μ*

_{*}, and the jet profile depends on both of these. We will not attempt to characterize this variation systematically. However, to make some contact with the strong-forcing limit considered by Farrell and Ioannou (2007), we consider the QL solution in Figs. 10b and 12a and increase the energy injection rate

*ε*by a factor of 1000, while holding

*β*,

*μ*and

*k*approximately fixed. Then from (49),

_{f}*β*

_{*}and

*μ*

_{*}are each reduced by a factor of 10. The time-averaged zonal mean profile of this strongly forced solution is shown in Fig. 12b and exhibits the parabolic velocity profile seen in the NL run in Fig. 1: there are fast eastward jets with sharp gradients and broad westward jets with smaller PV gradients. Also, the time-averaged QL jet profile in Fig. 12b is more asymmetric than the weaker forced QL jet shown in Fig. 12a, which has a forcing that is a factor of 10 smaller. To quantify the jet asymmetry, we use the ratio

*U*

_{max}and

*U*

_{min}are the maximum and minimum values attained in the zonal mean velocity profile. By increasing the forcing strength by a factor of 1000, the jet asymmetry increases from

*α*= 1.25 in Fig. 12a to

*α*= 1.56 for the profile in Fig. 12b. This is smaller than the “ideal,” marginally stable (i.e.,

*β*−

*U*= 0 everywhere except at the eastward jet where the PV jumps) profile considered in Danilov and Gurarie (2004), which has

_{yy}*α*= 2.

Comparison of time-averaged zonal mean velocity profiles (thin lines) of QL runs. (a) The solution from Fig. 10b with *β*_{*} = 1.0 and *μ*_{*} = 0.054; (b) the strongly forced solution with *β*_{*} = 0.1 and *μ*_{*} = 0.005. Also plotted are the corresponding PV gradients (thick curves).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of time-averaged zonal mean velocity profiles (thin lines) of QL runs. (a) The solution from Fig. 10b with *β*_{*} = 1.0 and *μ*_{*} = 0.054; (b) the strongly forced solution with *β*_{*} = 0.1 and *μ*_{*} = 0.005. Also plotted are the corresponding PV gradients (thick curves).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of time-averaged zonal mean velocity profiles (thin lines) of QL runs. (a) The solution from Fig. 10b with *β*_{*} = 1.0 and *μ*_{*} = 0.054; (b) the strongly forced solution with *β*_{*} = 0.1 and *μ*_{*} = 0.005. Also plotted are the corresponding PV gradients (thick curves).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Thus, although a detailed study of QL jet asymmetry is not a focus of the present work, our QL numerical solutions are generally consistent with the equilibrated SSST jets presented in Farrell and Ioannou (2007).

### e. Discussion of the eddy–eddy nonlinearity

An important effect of eddy–eddy nonlinearity is the stirring of PV, producing an exponential-in-time reduction in the length scale of vorticity fluctuations. Eddy-driven stirring is removed from the QL system by (9): shearing by *U*(*y*, *t*) is the only scale-reduction mechanism acting on the QL eddy vorticity. The small-scale structure evident in the QL PV gradient in the right panel of Fig. 11 may reflect the relative inefficiency of shearing by *U*(*y*, *t*) at removing vorticity fluctuations.

Further differences in the jet structure evident in Fig. 11 can be explained by meandering of the NL jets, so that the zonal average reduces the sharpness of the NL PV gradient. The spectral signature of the NL jet meanders is a high energy mode at *k _{x}*, is a well-known aspect of zonation. These are called a “satellite modes” by Danilov and Gurarie (2004), and they correspond to a domain-scale meander of the NL jets, which is not present in the QL case.

## 7. Discussion and conclusions

A contribution of this work is the analytic development of the linearized theory of zonostrophic instability within the context of the second-order cumulant expansion (CE2) of Marston et al. (2008) and the stochastic structural stability theory (SSST) of Farrell and Ioannou (2003, 2007). These statistical formulations are equivalent to the correlation dynamics derived in section 3, and that physical-space formulation, in terms of partial differential equations for the correlation functions Ψ and

In the top panel of Fig. 5 we display the curve of neutral zonostrophic stability in the (*β*_{*}, *μ*_{*})-parameter plane obtained by solution of linearized CE2 dynamics. We have shown that with isotropic forcing zonostrophic instability is not a negative-viscosity instability: the hallmark of a negative-viscosity instability is that at the stability boundary the most unstable wavenumber is zero. The deterministic model of anisotropically forced *β*-plane zonation analyzed by Manfroi and Young (1999) provides a bona fide example of the negative-viscosity case. Instead, for the isotropically and stochastically forced model analyzed here, the onset of zonostrophic instability is at the nonzero meridional wavenumber shown in the bottom panel of Fig. 5; only at large *β*_{*} does this wavenumber approach zero. Moreover, in section 5 we showed that with isotropic forcing the CE2 eddy viscosity *ν _{e}* is identically zero.

Comparison of QL and NL numerical solutions indicates that the CE2 linear stability boundary does not provide an accurate estimate of the onset of zonostrophic instability for NL flows. This quantitative failure of CE2 is not surprising: neglect of the eddy–eddy nonlinearity is most plausible in cases where most of the energy is in the zonal mean flow: close to the stability boundary the zonal mean flow is only incipient. An outstanding open problem is improving CE2 to account for the missing physics in the eddy–eddy nonlinearity. Another important problem is obtaining analytic insight into the solution of the CE2 system in the regime where CE2 is likely to be valid, namely in the strongly unstable regime where the drag *μ*_{*} is much less than the critical drag *μ ^{c}* and the fraction of energy in the zonal mean flow is substantial.

## Acknowledgments

This work was supported by the National Science Foundation under OCE1057838. We thank Nikos Bakas, Oliver Bühler, George Carnevale, Brian Farrell, Petros Ioannou, Brad Marston, Rick Salmon, and Steve Tobias for discussion of these results.

## APPENDIX A

### Implementation of the Random Forcing **ξ****(***x*, *t*)

**ξ**

*x*,

*t*)

*δ*-correlated forcing

*ξ*(

*x*,

*y*,

*t*) in (3) using a discrete approximation. The goal is to construct a statistically isotropic and narrowband forcing localized close to a radial wavenumber

*k*. Thus, the forcing is confined to an annulus

_{f}**k**in

*δk*=

*k*/8, so that

_{f}*k*. We use a fourth-order Runge–Kutta scheme, with time step

_{f}*δt*. Implementing the Runge–Kutta scheme requires the value of the forcing not just at points in time separated by the time step

*δt*but also at the midpoints. Some care must be exercised here, though, since the Runge–Kutta scheme assumes a certain degree of smoothness of the solution. To ensure this, we use a forcing that during the

*n*th time step, when (

*n*− 1)

*δt*<

*t*<

*nδt*, has the form

*n*th time step. The coefficient

*ξ*(

_{i}*n*) above is

*δ*correlated in the limit

*δt*→ 0. The phase

*φ*(

_{i}*n*) is a random variable, chosen from a uniform distribution in [0, 2

*π*]; the phase is set independently for each wave vector

**k**

_{i}and resets at the start of each time step.

^{b}(

*r*) and its idealized form, Ξ(

*r*) ∝

*J*

_{0}(

*k*) is shown in Fig. A1.

_{f}rComparison of the ring forcing *δk* → 0 (solid line) and the narrow-band forcing with *δk* = *k _{f}*/4 (dashed line).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of the ring forcing *δk* → 0 (solid line) and the narrow-band forcing with *δk* = *k _{f}*/4 (dashed line).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

Comparison of the ring forcing *δk* → 0 (solid line) and the narrow-band forcing with *δk* = *k _{f}*/4 (dashed line).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0200.1

## APPENDIX B

### Rapid Temporal Decorrelation: Derivation of (25)

*δ*(

*t*

_{1}−

*t*

_{2}) on the right-hand side of (14). Operationally, this means that we might integrate (11) during the first time step from

*t*= 0 to

*t*=

*δt*as

**x**, 0) indicates “all other terms” in (11), evaluated at

*t*= 0. Also in (B1),

*n*th time step one creates a new independent realization of

*δt*as

*δt*→ 0, and therefore

*δt*→ 0. As demanded by this argument, notice that

*ξ*(

_{i}*n*) in (A3) is proportional to

**x**

_{1}with (B1) evaluated at

**x**

_{2}to obtain

*n*indicates evaluation at

**x**

_{n}; for example,

*ζ*′(

**x**, 0). Thus,

*δt*→ 0 the left-hand side is the time derivative of the vorticity correlation function. The

*δt*→ 0 in (B5), we obtain the deterministic differential equation (23) for the evolution of the correlation function

## APPENDIX C

### Derivation of the Dispersion Relation (45)

*U*(

*y*,

*t*) in (44) one has

_{+}(

*s*′,

*m*) and Λ

_{−}(

*s*′,

*m*) are defined by the integral

*p*→ −

*p*and

*q*→ −

*q*, and using the exchange symmetry in (22), one finds that

*m*Λ

_{−}. Then with the change of variables

*q*′ =

*q*−

*m*/2 in the Λ

_{−}integral, and using (41), one can write the dispersion relation (C13) as

*p*,

*q*) =

*h*(cos

*θ*, sin

*θ*):

*S*is the function

*S*(

*χ*,

*n*) = −

*S*(−

*χ*,

*n*) = −

*S*(

*χ*, −

*n*), and therefore

*S*(0,

*n*) =

*S*(

*χ*, 0) = 0. These symmetries are important for further work, and they are not manifest from the definition of

*S*in (C18). Thus we seek an alternative form with more obvious properties. The change of variables

*θ*→

*θ*+

*π*results in

*Q*(

*χ*,

*n*) is manifestly an even function of

*n*, and

*θ*→ −

*θ*shows that

*Q*is also an even function of

*χ*.

## APPENDIX D

### The Function *Q*(**χ****,** **n****)**

**χ**

**n**

In this appendix we summarize some properties of the function *Q*(*χ*, *n*) defined in (C20).

*n*

^{2}≤ 1 then

*β*

_{*}≪ 1 requires the approximation of

*Q*(

*χ*,

*n*) in the limit

*χ*→ ∞. One can expand the integrand in inverse powers of

*χ*and integrate term by term. The first two nonzero terms are

*β*

_{*}≫ 1 requires the approximation of

*Q*(

*χ*,

*n*) in the limit

*χ*→ 0. A somewhat laborious “range-splitting” calculation shows that

## APPENDIX E

### The Neutral Curve

*β*

_{*},

*μ*

_{*})-parameter plane is defined by the conditions in (53). For the dispersion relation in (51), these take the form

*χ*

_{0}= 2

*μ*/

*mβ*above. (For brevity, in this appendix, we drop the asterisks indicating nondimensional variables.) An examination of the numerical values of

*χ*

_{0}on the neutral curve motivates the possibility that

*χ*

_{0}→ ∞ as

*β*→ 0 and

*χ*

_{0}→ 0 as

*β*→ ∞. We now use this numerical observation to derive analytical approximations for the neutral curve in the complementary limiting cases

*β*→ 0 and

*β*→ ∞. To this end, we use the approximations to

*Q*(

*χ*,

*m*) summarized in appendix D.

#### a. Approximation of the marginal curve, β ≫ 1 and χ_{0} ≪ 1

*β*→ ∞, we have from (D4)

*O*(

*mχ*

_{0}) term must be justified post facto once a consistent dominant balance is found. Substituting (E3) in the neutral curve equations, (E1) and (E2) and keeping in mind that that

*β*

^{−1}≪ 1, we get

*m*equation corresponds to

*m*

^{3}~ 3

*β*

^{−3}and, consequently,

*m*~

*O*(

*β*

^{−1}) the only consistent balance in (E1) is

*μ*~ 2

*β*

^{−2}. A higher-order estimate for

*μ*can be derived by substituting for

*m*to get

*β*> 2.

#### b. Approximation of the marginal curve, β ≪ 1 and χ_{0} ≫ 1

*β*→ 0 for which we can write, from (D3),

*μ*from (E12), gives us

*m*~ 1 and

*m*against the numerical estimate in Fig. 5 and the agreement is excellent. In fact, the approximations to

*m*(

^{c}*β*) practically span the entire range of the neutral curve.

#### c. The small-m expansion

*m*→ 0 with all other parameters fixed, we observe in Fig. 4 that

*s*→ −

*μ*and therefore in (51) the first argument of

*Q*is

*m*expansion of the dispersion relation we write

*s*= −

*μ*+

*s*

_{1}and use (D5) to approximate

*Q*. Thus the dispersion relation (51) becomes

*m*version of the dispersion relation in (59).

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^{1}

There is also uniform advection by the zonal flow. But that sweeping is eliminated by the difference *U*_{1} − *U*_{2} in the correlation equation (31) and is therefore inconsequential to Reynolds stresses.

^{3}

In certain cases the system may be transitioning between a state with *n* and *n* + 1 jets. Following Panetta (1993), we then count

^{4}

From (3), the exact energy power integral is 〈*ψξ* + *μ*|**∇***ψ*|^{2} + (−1)^{n−1}*ν _{n}*|

**∇**

^{n−1}

*ζ*|

^{2}〉 = 0, where 〈〉 is both a domain integral and a time average.

^{5}

If *β* = 0 then the equations of motion are invariant under *y* → −*y* and *ψ* → −*ψ*. This symmetry, which induces *u* → *u*, is broken in both QL and NL by nonzero *β*. This explains the characteristic east–west asymmetry of *U*(*y*, *t*) on the *β* plane.

^{6}

Farrell and Ioannou (2007) also employ a forcing with a different correlation function than our isotropic choice.