a. Quasilinear dynamics

The main results in this paper are obtained with a quasilinear (QL) system, which is defined by taking

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in (7). Figure 2 shows a QL solution at the same parameter values as the fully nonlinear (NL) solution as Fig. 1. Because of the coupling between the mean and the eddies, the QL system is nonlinear, and Fig. 2 shows that QL dynamics still results in the spontaneous formation of quasi-steady zonal jets.

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_fL = 32). The snapshot is at 2μt = 40 after spinup from rest.

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

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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_fL = 32). The snapshot is at 2μt = 40 after spinup from rest.

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

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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_fL = 32). The snapshot is at 2μt = 40 after spinup from rest.

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

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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

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(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

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(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

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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

A main motivation of the QL system is that using the statistical method pioneered in meteorology by Farrell and Ioannou (1993b, 2003, 2007) one can compute important average quadratic properties of the QL flow, such as Reynolds stress and the eddy enstrophy and energy. However, rather than (9), Farrell and Ioannou (2007) adopt a stochastic closure, which amounts to replacing the eddy–eddy nonlinearity with a combination of random forcing and dissipation:

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(see also DelSole 2001). The intent of (10) is that the random forcing ξ_EENL(x, t) and the dissipation μ_EENLζ′ 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.

a. Correlation functions: Kinematics

We assume that the external forcing, a(x, t) in (1) and ξ(x, t) in (11), has two-point, two-time correlation functions of the form

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where the overbar above denotes an ensemble average. The dependence of the spatial correlation functions A and Ξ only on the difference

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indicates that the forcing is zonally homogeneous. We do not assume (yet) that the forcing is isotropic and meridionally homogenous.

Because derivatives commute with the ensemble average, relation (4) implies that A and Ξ are related by

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where the Laplacian acting on the coordinates of point n is

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Notice that that we have changed notation: undecorated x in (15) is the zonal difference coordinate. We also use the shorthand , etc. Strictly speaking, we should decorate all the variables in (11) with the subscript n = 1 or 2 to explicitly indicate whether we refer to the eddy vorticity equation at the point x₁ = (x₁, y₁) or at the point x₂ = (x₂, y₂). We forbear from doing so.

We assume “ergodicity” so the overbar is also equivalent to the zonal average of a single realization. We desire the single-time two-point correlation functions

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and

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The analog of (16) is

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Given the streamfunction correlation Ψ(x, y₁, y₂, t), one can obtain the velocity correlation tensor as

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Because the choice of denoting one point as x₁ and the other as x₂ is arbitrary, all correlation function have an important “exchange” symmetry

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and likewise for A, , Ξ, etc.

b. Correlation functions: Dynamics

It follows from (11) and (14) that the correlation functions evolve as

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where the Rayleigh–Kuo operator at point n is .

To derive (23), multiply the equation for by and add this to the equation multiplied by . The sum is then ensemble averaged, and after the average all fields depend on x₁ and x₂ only through the combination x = x₁ − x₂. Because of this zonal homogeneity . Thus, for example,

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A crucial simplification is that the forcing is rapidly decorrelating in time, as expressed by the δ(t₁ − t₂) in (14). Considerations summarized in appendix B (amounting to a simple proof of Ito’s formula) show that

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The result above is the origin of the first term on the right-hand side of (23).

c. Collective coordinates

As alternatives to y₁ and y₂, there are advantages in using the “collective coordinates”

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Eventually we will restrict attention to homogenous and isotropic forcing, and at that point we take Ξ in (14) to be a function only of the two-point separation

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Collective coordinates are then essential for analytic progress.

In terms of y and , the Laplacians are

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where is the “separation” Laplacian. Thus, for instance,

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Using the coordinates in (26), the correlation equation (23) becomes

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where now and and .

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 then the eddy enstrophy is .

A key result, obtained by evaluating

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at (x, y) = 0, is that the Reynolds stress is

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Thus, the mean flow equation (6) can be written as

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The mean flow equation (34), coupled with the correlation equation (31), is a closed system for the ensemble-averaged properties of QL dynamics.

a. The spatially homogeneous basic state

We now suppose that the forcing is statistically homogenous and isotropic, namely that the correlation function of the forcing has the particular form

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where r is the two-point separation defined in (27). Because Ξ does not depend on , there is a simple exact solution to (31) and (34). This solution is spatially homogeneous and isotropic and has no mean flow, U = 0. With these simplifications the correlation equation (31) collapses to

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The subscript H emphasizes that is spatially homogeneous (i.e., independent of ). The streamfunction correlation function Ψ_H(x, y) can be obtained from by solving . It is remarkable that in (36) is independent of β: an isotropic and spatially homogenous forcing drives an isotropic and spatially homogeneous flow, despite the anisotropy of Rossby wave propagation.

We now apply the Fourier integral theorem,

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to (36). We use the notation

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so that after the Fourier transform , and the streamfunction spectrum is related to the forcing spectrum by

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We emphasize that in (40) is not singular as h → 0. To see this, we recall (16), which in this homogeneous and isotropic case implies that Ξ = ∇⁴A, and therefore . In terms of then, the streamfunction spectrum in (40) becomes

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Since a(x, t) is stationary, the spectra and are finite as h → 0 (provided that μ ≠ 0).

Later we will need two integral constraints on the vorticity forcing correlation function Ξ:

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and also

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These follow from Ξ = ∇⁴A, 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

The linear stability of the spatially homogeneous solution in (41) is determined by imposing small initial disturbances and examining evolution in time. The perturbation variables are added to the base-state variables, (0, Ψ_H, ) and substituted into (31) and (34). The total “flow,” with mean and imposed small perturbations, is

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where 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 , one has the equations governing the evolution of small perturbations to the homogeneous solution. The details of the subsequent solution are in appendix C, and a main result of that analysis is the dispersion relation

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where is the forcing spectrum in (41), and the function Q(χ, n) is defined by the angular integral

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The dispersion relation (45) applies to the special case of isotropic forcing, 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 , which Carnevale and Martin (1982) consider to be fourth order in wave amplitude and therefore negligible. However, after linearization of CE2 around a basic state with 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

In most previous investigations of zonation, the forcing is limited to an annulus of wavenumbers in Fourier space. Typically the annulus of forced modes has a mean radius h = k_f and has thickness 2δk ≪ k_f. This is the “narrow-band forcing” described in appendix A. We idealize this choice further by considering “ring” forcing corresponding to the limit δk → 0. In other words, we consider a random flow, driven isotropically by injecting energy on the circle h = k_f in wavenumber space. This corresponds to

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where J₀ is the Bessel function of order zero. Notice that . With ν = 0—as we assume in (45)—the spatially homogeneous base-state solution in (41) is

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The parameter 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.

With δ(h − k_f) in (47), the 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 and time scale . These scales lead to the control parameters

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The nondimensional wavenumber and growth rate are

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The zonostrophic dispersion relation in nondimensional variables is then

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with the function 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_i. This numerical solution indicates that modes with s_r > 0 are found only if 0 < m² < 1, and these unstable modes have s_i = 0. We have been unable to obtain a satisfactory nonnumerical proof of these two important properties of zonostrophic instability.

Figure 4 shows some examples of the growth rate 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:

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A main conclusion resulting from our analysis is that zonostrophic instability is suppressed if 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

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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

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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).

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The marginal stability condition in (52), which is equivalent to the requirements

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defines a “critical curve” in the (β_*, μ_*) parameter plane. This curve, is shown in the upper panel of Fig. 5. The solution is linearly unstable in the region below the critical curve. The peak of the critical curve is . This peak defines the “most unstable” point in the (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 (solid line); linear zonostrophic instability occurs in the region below the critical curve. (b) The wavenumber on the critical curve (solid line; i.e., the most linearly unstable wavenumber). Asymptotic approximations for the critical curve and the most unstable wavenumber based on β_* ≫ 1 (dash-dotted) and β_* ≪ 1 (dashed) are shown in both panels.

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(a) The critical curve (solid line); linear zonostrophic instability occurs in the region below the critical curve. (b) The wavenumber on the critical curve (solid line; i.e., the most linearly unstable wavenumber). Asymptotic approximations for the critical curve and the most unstable wavenumber based on β_* ≫ 1 (dash-dotted) and β_* ≪ 1 (dashed) are shown in both panels.

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(a) The critical curve (solid line); linear zonostrophic instability occurs in the region below the critical curve. (b) The wavenumber on the critical curve (solid line; i.e., the most linearly unstable wavenumber). Asymptotic approximations for the critical curve and the most unstable wavenumber based on β_* ≫ 1 (dash-dotted) and β_* ≪ 1 (dashed) are shown in both panels.

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

Also shown in Fig. 5 are analytic approximations in the complementary limits 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

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with wavenumber

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In the complementary limit β_* ≫ 1, the approximation to the critical curve is

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with wavenumber

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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

The structure of the dispersion relation at small m provides insight into the nature of zonostrophic instability. Looking at Fig. 4, we anticipate that

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where η₂ > 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).

However there is a small surprise: from (E17) we find that the expansion of the dispersion relation (51) around m = 0 is

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That is, the term η₂ 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

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leading to the small-m growth rate in (59). We analyze this curious situation further in section 5 and show that η₂ = 0 follows from the assumed isotropy of the forcing [i.e., η₂ = 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 η₂, and that the hyperviscous coefficient η₄ 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.

a. A solution of the correlation equation

We can simplify (61) with (i.e., by looking for a solution independent of ):

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This exact reduction is surprising because the 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

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with the initial condition

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The solution of the steady problem (63) is then obtained as

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Thus, solving the initial-value problem for F, the vorticity correlation function is written as the time integral of a sheared disturbance:

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b. The Reynolds stresses

To obtain the correlation function Ψ from we must solve the two-dimensional biharmonic equation , which is accomplished with the Green’s function defined by ∇⁴G = δ(x)δ(y), or , or

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With G(x, y) in hand, we have

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The Reynolds stress now follows by evaluating Ψ_xy at zero separation, or

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This is a very convenient and general expression for the Reynolds stresses in terms of the vorticity correlation function .

Substituting (67) into (70) results in a triple integral. To disentangle this, exchange the order so that the t integral is last, and in the inner x and y integrals “unshear” the correlation function with the coordinate change:

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After these maneuvers the Reynolds stress is

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where

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Now we restrict attention to isotropic forcing; that is,

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Then in polar coordinates, the double integral in (73) factors:

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The final line follows from constraint (42) and implies that . That is, the eddy viscosity is zero.

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 , which follows from the integral constraints on Ξ in (42) and (43).

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

The energy power integral for the β-plane Couette flow problem considered here is obtained by first rewriting (63) as

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Canceling a Laplacian, and evaluating the result at zero separation, one obtains²

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The left-hand side is the transfer of energy between the eddies and the Couette flow, which is zero because Therefore the statistically energy balance is between dissipation due to drag and the rate of working of the random force that drives the eddies. Remarkably, because the Reynolds stresses are zero, the eddy kinetic energy of the statistically steady flow is ε/(2μ), 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_e, 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.

a. The onset of zonation in NL and QL solutions

Shown in the bottom panel of Fig. 3 is the evolution of the fraction of kinetic energy in the zonal mean flow

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where denotes the area integral over the entire domain. The index zmf(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 t₁ < t < t₁ + 10/μ, where typically 2μt₁ > 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 correspond to the critical curve in the upper panel of Fig. 5; these analytic predictions compare well with the increase in 〈zmf〉 in the QL numerical solutions. The dotted lines marked in Fig. 6 are eyeball estimates of the onset of NL zonation.

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

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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

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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.

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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 is as large as 5. Thus the QL system is much more unstable than the NL system. Regarding this quantitative difference between NL and QL, we recall that QL (and CE2) is an approximation based on dropping the eddy–eddy nonlinearity. This approximation is most defensible when the mean flow is very strong (i.e., in cases where the zonal mean flow contains almost all of the energy). Therefore, CE2 is not likely to be quantitatively accurate near the linear stability boundary, where the zonal mean flow is weak or nonexistent. The comparison in Fig. 6 is thus a worst-case test of CE2. How, or if, CE2 might be improved to account for the missing eddy–eddy nonlinearity in this weak mean-flow regime is an open issue.

b. Zonostrophically stable NL solutions

Figure 7 shows two NL solutions that are zonostrophically stable; that is, these solutions have

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and 〈zmf〉 ≈ 0. In the left panel of Fig. 7 the drag is so heavy that the approximate dominant balance in (1) is 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 2μt = 25, after spinup from rest, and β_* = 1.

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

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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 2μt = 25, after spinup from rest, and β_* = 1.

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

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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 2μt = 25, after spinup from rest, and β_* = 1.

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

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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_f) for QL and NL solutions with β_* = 1. (b),(d) The residual spectrum E_R(k/k_f), defined as the angularly averaged spectrum after removal of the “zonal modes” with k_x = 0. The largest peak in E_Z(k_y/k_f) defines the wavenumber m_Z, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with μ_* = 0.0545 in (a)].

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

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(a),(c) The zonal spectrum E_Z(k_y/k_f) for QL and NL solutions with β_* = 1. (b),(d) The residual spectrum E_R(k/k_f), defined as the angularly averaged spectrum after removal of the “zonal modes” with k_x = 0. The largest peak in E_Z(k_y/k_f) defines the wavenumber m_Z, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with μ_* = 0.0545 in (a)].

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

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(a),(c) The zonal spectrum E_Z(k_y/k_f) for QL and NL solutions with β_* = 1. (b),(d) The residual spectrum E_R(k/k_f), defined as the angularly averaged spectrum after removal of the “zonal modes” with k_x = 0. The largest peak in E_Z(k_y/k_f) defines the wavenumber m_Z, even if there are no quasi-steady zonal jets [e.g., as in the NL simulation with μ_* = 0.0545 in (a)].

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

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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 range. However this solution has 〈zmf〉 ≈ 0 and thus serves as example of an isotropic, spatially homogeneous, weakly turbulent, 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 estimate³ a jet wavenumber m_J. For example, in Fig. 1 there are seven jets and therefore .

However we noticed that there are cases without jets in which the zonal energy spectrum E_Z(k_y/k_f) has a strong peak: an example is the 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_Z that is the peak of the zonal spectrum E_Z(k_y/k_f). In cases where there are strong jets we invariably find that m_Z ≈ m_J. It is interesting to compare m_J and m_Z with a Rhines wavenumber defined as

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where the root-mean-square velocity is

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We investigated other choices for the velocity in the Rhines wavenumber; for example, Rhines (1975) advocated the RMS of 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 integral⁴ can be used to express V_RMS in terms of external parameters as

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The relation above applies with an error (due to hyperviscous dissipation) of less than 5% in our simulations. Substituting (82) into (80) one has

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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

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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

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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

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At large values μ_* only the directly forced modes are excited, and consequently m_Z ≈ k_f in both the QL and NL cases. At the critical value in Fig. 9, the QL solutions undergo zonostrophic instability, and close to this transition (e.g., at μ_* = 0.2 and 0.165 in Fig. 9a) the QL m_Z 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 m_Z begins to decrease.

There is an interesting transition at in Fig. 9. At this point the QL and NL zonal wavenumbers are equal, and as μ_* is reduced the QL and NL wavenumbers are locked together. At in Fig. 9 the NL solutions finally become zonostrophically unstable, resulting in NL jets and significantly nonzero values of NL 〈zmf〉. At the smallest value of 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 In the strongly unstable regime, with μ_* significantly less than the observed wavenumber is much smaller than the most unstable wavenumber predicted by linear theory. This increase in the QL jet scale is a result of merging jets that initially appear with a spacing, which is well predicted by the linear theory. This phenomenology begins at about and is illustrated in Fig. 10.

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

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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

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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

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

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

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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

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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

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Via integration of their SSST system, Farrell and Ioannou (2007) report equilibrated zonal mean flows with much stronger east–west asymmetry than the QL flow in Fig. 11 (e.g., see their Figs. 8 and 9). There are at least⁶ 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_f approximately fixed. Then from (49), 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

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

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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

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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

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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 in the two-dimensional NL spectrum; this same mode is only weakly excited in the QL spectrum. The excitation of almost-zonal modes, with a small but nonzero of value of 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.