## 1. Introduction

Coherent jets are often observed in turbulent flows, with the banded winds of Jupiter constituting a familiar and frequently studied example (Ingersoll 1990; Vasavada and Showman 2005; Sánchez-Lavega et al. 2008). This phenomenon of spontaneous jet formation in turbulence has been studied observationally and theoretically (Rhines 1975; Williams 1979, 2003; Panetta 1993; Nozawa and Yoden 1997; Huang and Robinson 1998; Manfroi and Young 1999; Vallis and Maltrud 1993; Cho and Polvani 1996; Galperin et al. 2004; Lee 2005; Kaspi and Flierl 2007; Sokolov and Rintoul 2007), as well as in laboratory experiments (Krishnamurti and Howard 1981; Read et al. 2004, 2007). The primary mechanism maintaining planetary turbulent jets has been identified as eddy momentum flux systematically directed up the mean velocity gradient (Panetta 1993; Nozawa and Yoden 1997; Huang and Robinson 1998; Ingersoll et al. 2004; Salyk et al. 2006). This upgradient momentum flux is produced by shear straining of the eddy field by the mean flow, implying that the large-scale jets are maintained by a spectrally nonlocal interaction between the small-scale eddy field and the large-scale jets (Panetta 1993; Nozawa and Yoden 1997; Huang and Robinson 1998; Ingersoll et al. 2004; Salyk et al. 2006; Kitamura and Ishioka 2007).

Characteristics of the jets of Jupiter include prograde equatorial velocity, multiple midlatitude jets, asymmetry between prograde and retrograde jets such that the prograde jet is typically stronger and sharper than the retrograde jets, and substantial change in sign of the absolute vorticity gradient in the midlatitude retrograde jets (Limaye 1986; Porco et al. 2003; Read et al. 2006).

The central component of a theory for turbulent jet dynamics, that is, the method for obtaining the structure of the turbulence and the associated fluxes given the jet, is provided by a stochastic turbulence model (STM) (Farrell and Ioannou 1993a, 1994, 1995, 1996; DelSole and Farrell 1995; DelSole 1996; DelSole and Farrell 1996; Newman et al. 1997; Whitaker and Sardeshmukh 1998; Zhang and Held 1999; DelSole 2004). Once these fluxes are known, the equilibrium jet dynamical balance between the ensemble-mean turbulent momentum flux convergence and dissipation can be determined. This is the method of Stochastic Structural Stability Theory (SSST; Farrell and Ioannou 2003, 2008). The interaction between the large-scale jet structure and the field of eddy turbulence is nonlinear and results in a nonlinear trajectory for the jet and the ensemble-mean eddy field associated with it, which for parameter values in the shallow-water equations (SWEs) on an equatorial beta plane typical of Jovian conditions asymptotically approaches a stable equilibrium with multiple jets. This equilibrium arises from a linear instability of the coupled turbulence–mean flow system. This jet-forming instability is emergent from the interaction between the mean flow and the turbulence. The finite-amplitude equilibria can, for sufficiently low turbulence intensity, be connected continuously to the associated linear instabilities.

In this work, the convective excitation of turbulence observed in Jupiter’s atmosphere (Gierasch et al. 2000; Ingersoll et al. 2000) is approximated as stochastic in both space and time and its amplitude taken as an independent parameter in the solution of the SSST system. Using this model we investigate both the formation and equilibration of jets in SWE equatorial beta-plane turbulence.

We find that if the turbulence is not too strong, and if a randomly structured small-amplitude initial zonal jet is prescribed, the equatorial jet equilibrium can be either prograde or retrograde, depending on the initial perturbation zonal velocity. This result is consistent with the existence of linear instability of the zonal jet free turbulent state to zonal jet perturbations that occurs in the presence of sufficient turbulence intensity. However, for parameter values used in our model simulations, as the turbulence level increases or the dissipation of the jet decreases, the nonlinear equilibrium equatorial jet speed increases and the retrograde equatorial jet ultimately loses structural stability. At this point the retrograde jet undergoes a catastrophic readjustment, from which it emerges as a prograde jet. On the other hand, the prograde equatorial jet persists as a stable equilibrium at higher turbulence intensity and/or lower jet dissipation, with correspondingly higher associated jet speed. The stability of both prograde and retrograde jets over a considerable parameter range is consistent with the mix of prograde and retrograde jets on the gaseous planets and in simulations of decaying turbulence initialized randomly (Kitamura and Ishioka 2007), as well as in simulations of forced turbulence (Huang and Robinson 1998; Scott and Polvani 2007, 2008). The phenomenon of retrograde jet destabilization occurs over a range of parameter values typical of Jupiter jet simulations but has not been verified to occur for all parameter values appropriate for the gaseous planets, particularly for Froude numbers of *O*(1) or larger.

At low turbulence intensity, multiple nearly symmetric prograde and retrograde jets with low jet speed form in the midlatitudes. As turbulence intensity increases, the jet speed also increases, with the prograde jets becoming increasingly narrow while the retrograde jets assume a more rounded profile. These adjustments occur because as turbulence intensity rises, the retrograde jet encroaches on eddy stability boundaries, which results in modification of the jet structure by eddy momentum fluxes because these eddy fluxes diverge at the stability boundary (Farrell and Ioannou 2003). The stable equilibrium jets exhibit a substantial reversal in sign of both absolute and potential vorticity gradients. Such meridionally localized reversals of absolute vorticity gradient are a prominent feature of both observed and simulated jets (Limaye 1986; Read et al. 2006; Scott and Polvani 2007). Potential vorticity gradients, reversals of which are associated with necessary conditions for instability in the inviscid SWE system, change sign in approximate coincidence with absolute vorticity gradients in our jets, in observations, and in simulations (Read et al. 2006; Scott and Polvani 2007). We find that our jets remain stable with a limited extent of reversal of potential vorticity gradient, which implies, given the approximate coincidence of absolute and potential vorticity gradients, that jet spacing is close to the notional Rhines radius based on the mean velocity.^{1} However, this meridional jet scale arises in SSST from a mechanism unrelated to the Rhines radius interpreted as a halting scale for two-dimensional turbulent cascades.

## 2. Dynamics of zonal jets in turbulence

A theory of zonal jet dynamics in turbulence was developed in Farrell and Ioannou (2003). This theory was applied to the problem of the formation of jets in barotropic turbulence in Farrell and Ioannou (2007) and to the problem of formation of jets in baroclinic turbulence in Farrell and Ioannou (2008). We now briefly review this theory in the context of the SWE.

*x*) velocity,

*u*, meridional (

*y*) velocity,

*υ*, and height,

*h*, areThe Coriolis parameter for the equatorial beta plane is

*f*(

*y*) =

*βy*with

*β*= 2

*ω*/

*a*, where

*ω*is the rotation rate of the planet and

*a*is its radius. We will also consider the geophysical beta variation

*f*(

*y*) = (

*βL*/

_{y}*π*) sin(

*πy*/

*L*) over the channel interval [−

_{y}*L*/2,

_{y}*L*/2]. Rayleigh friction at rate

_{y}*r*is included in the momentum equations and Newtonian cooling, at the rate

_{u}*r*, in the height equation. We also include a latitudinally varying Rayleigh friction

_{h}*r*(

*y*):that damps all fields, creating an absorbing sponge layer in the vicinity of the channel walls. Diffusive dissipation with coefficient

*ν*is added to improve numerical stability.

Height is nondimensionalized by the equivalent depth of the fluid, *H*; the horizontal scales by a typical length scale, *L*; time by the period of rotation of the planet, *T* = 2*π*/*ω*; and velocity by *U*_{0} = *L*/*T*. The Froude number is *F* = *U*_{0}/*gH**g* being the gravitational acceleration. For Jupiter *T* = 3.6 × 10^{4} s, and with *L* = 10^{7} m, the velocity scale is *U*_{0} = 278 m s^{−1} and the nondimensional equatorial beta parameter is *β* = 1.8. In most of the examples *F* = 0.3, *r _{s}* = 2,

*δ*= 0.5, and

_{s}*ν*= (6

*δy*)

^{2}, where

*δy*is the discretization interval in the meridional direction.

For the Jovian case we assume the fluid is forced by convection and is in a turbulent state. To model forcing of the mean jets by this turbulence, we use a stochastic turbulence model to obtain the ensemble mean eddy fluxes. The stochastic forcing and the damping parameters for Jupiter are not well known individually. What is known is the turbulent large-scale rms velocity, which is *O*(5 m s^{−1}), and the mean jet velocity, which is *O*(100 m s^{−1}) (Salyk et al. 2006). By adjusting the forcing and eddy damping to produce the observed level of turbulence, one can infer the linear mean jet dissipation rate required to produce the observed jet velocity. Combining the Lyapunov equation of the STM with the equation for the zonal mean flow, we obtain the autonomous deterministic equation set from which the jets in equilibrium with the turbulence can be determined (Farrell and Ioannou 2003). The development of these equations is now described in more detail.

^{2}

*D*denotes differentiation with respect to

*y*and 1/

*r*is the damping time of zonally averaged fields, which will be taken to be on the order of the radiative time scale 1/

_{m}*r*≈ 1000 Jovian days. The mean height

_{m}*h*

*F*(

_{u}η_{u}*t*),

*F*(

_{υ}η_{υ}*t*), and

*F*(

_{h}η_{h}*t*) parameterize the structure and time variation of the convective excitation and the excitation from the nonlinear eddy–eddy scattering (cf. Farrell and Ioannou 2003). Because of zonal homogeneity, the fields can be expanded in a Fourier series with zonal wavenumber

*κ*= 2

*πm*/

*L*, where

_{x}*L*is the equatorial circumference and

_{x}*m*is the global zonal wavenumber. The perturbation field at each wavenumber evolves according toin which the state,

**, has been discretized on an equally spaced grid to form the state column vector: [**

*ψ**u*

_{1}, … ,

*u*,

_{n}*υ*

_{1}, … ,

*υ*,

_{n}*h*

_{1}, … ,

*h*]

_{n}^{T}, with subscript

*i*denoting the value of the field at the

*i*th discretization point and

*n*the number of points. The associated dynamical operator 𝗔

_{κ}(

*u*

*ϵ*controls the excitation amplitude.

*f*, so large divergent velocity fields are excited and a computationally expensive geostrophic adjustment needs to be resolved. To avoid this, the excitation is assumed to occur on space and time scales that excite nearly balanced velocity and height fields. We choose each column of the

*n*columns of the

_{f}*n*×

*n*excitation matrix, 𝗙, to excite the rotational component of the velocity by exciting the streamfunction proportional to exp[−(

_{f}*y*−

*y*)

_{i}^{2}/

*δ*

_{f}^{2}], where

*y*is the

_{i}*y*coordinate at the

*i*th collocation point and

*δ*= 1, resulting in spatial correlation of the excitation over

_{f}*L*= 10

^{6}m (this excitation matrix is assumed to be the same for all zonal wavenumbers, and we find it produces an approximately flat perturbation energy spectrum over the dynamically important zonal wavenumbers

*m*= 5–60). A red noise process is used to correlate the above excitation structures in time. This red noise process is implemented by choosing the

*n*red noise process

_{f}**(**

*η**t*) to solve the equationwhere

*τ*is the autocorrelation time scale of the red noise process and

**w**is a white noise process with zero mean and unit variance:in which 𝗜 is the identity matrix, 〈·〉 denotes an ensemble average, and † denotes the Hermitian response. The factor

*τ*

*κ*, evolves according to the time-dependent Lyapunov equation (Farrell and Ioannou 1996)in whichis the operator of the perturbation dynamics augmented by the red noise process. The associated excitation is

*κ*:

*is the operator that projects the state,*

_{u}**, on the complex Fourier amplitude of its zonal velocity,**

*ψ**u*, and 𝗣

*is the corresponding projector on the amplitude of the meridional velocity,*

_{υ}*υ*, we then havewhere ℜ denotes the real part, * denotes complex conjugation, and diag is the operation that forms a diagonal matrix with the elements of the column vector on which it operates or forms a vector from the diagonal of the matrix on which it operates.

**u**

^{κ}and

*u*

*u*^{κ}depends linearly on

*u*^{κ}. This nonlinear system is globally stable and its typical temporal asymptotic solutions are steady jets; although limit cycle and chaotic trajectories are supported for some parameter values (Farrell and Ioannou 2003).

## 3. Mechanism of jet formation

*ϵ*, eddy–mean flow interaction is inadequate to overcome dissipation, jet growth does not occur, and the mean flow remains constant and close to zero. We call this nearly motionless state the zero state. The stability of the zero state can be studied by perturbing the SSST system [(13) and (18)] and investigating the eigenvalues of the corresponding coupled perturbation equations for

*δ*

**C**, the perturbation covariance, and

*δ*

*u*_{0}

^{κ}is the eddy covariance maintained at equilibrium by unit excitation

*ϵ*= 1 when the background flow is the zero state with associated linear operator

*δ*𝗖

^{κ}, linearly on

*δ*

*u**δ*

*u**ϵ*(because the covariance at equilibrium is a linear function of the excitation level). For a critical value of the excitation

*ϵ*, the SSST perturbation system becomes unstable, and for values larger than the critical a new mean flow arises, which for excitations close to

_{c}*ϵ*has nearly the same structure as the most unstable eigenfunction of the SSST perturbation equations.

_{c}*ϵ*and the structure of the eigenfunctions (denoted with the subscript

_{c}*eig*)at the bifurcating point can be obtained with economy when at the bifurcating point the top eigenvalue of (19) and (20) is real and zero (i.e., when the principle of exchange of stability holds). When this is true, the structure of the instability at the bifurcating point can be studied very simply because at

*ϵ*Eq. (20) implies that the eigenfunction

_{c}*δ*

*u*_{eig}and

*ϵ*and the perturbation eddy-induced acceleration in (19) can be written aswhere 𝗠 is a linear operator. Substituting (23) in (19), we obtain in the absence of sponge layers and diffusive damping that

_{c}*δ*

*u*_{eig}is an eigenfunction of 𝗠:The critical destabilizing

*ϵ*for an eigenfunction is given by

_{c}*ϵ*=

_{c}*r*/

_{m}*λ*, where

*λ*> 0 is a corresponding positive eigenvalue of 𝗠.

*ϵ*= 1) to small changes in the zonal mean wind; specifically, we interpret −〈(

_{c}*υ*′

*Du*′)〉

*to be, for unit excitation, the eddy acceleration induced at the*

_{i}*i*th collocation point by a zonal mean wind perturbation at the

*j*th collocation point,

*u*

_{j}.

For simplicity assume an eddy field restricted to have a single perturbation zonal wavenumber, *m*, and obtain the critical *ϵ _{c}* and the corresponding top eigenfunction of

*ϵ*𝗠 −

_{c}*r*𝗜 as a function of

_{m}*m*. The eigenfunction of

*ϵ*𝗠 −

_{c}*r*𝗜 with zero growth gives the mean flow structure that will emerge at small but supercritical

_{m}*ϵ*>

*ϵ*. At the associated

_{c}*ϵ*the zero state becomes structurally unstable and jets emerge from this monochromatic eddy field. The critical rms turbulence intensity

_{c}*r*/

_{m}*λ*, where

_{m}*λ*is the top positive eigenvalue of 𝗠, is shown as a function of zonal wavenumber

_{m}*m*in Fig. 1. The critical excitation

*ϵ*decreases rapidly with eddy perturbation zonal wavenumber

_{c}*m*, implying that if the high eddy perturbation zonal wavenumbers are not suppressed, the flow will be organized (at least for sufficiently small zonal wind perturbations) by these high-zonal-wavenumber eddies. Such an increase in forcing of the mean jet with increase in zonal perturbation wavenumber was noted by Kitamura and Ishioka (2007).

The top eigenfunctions of *ϵ _{c}*𝗠 −

*r*𝗜 for eddy zonal wavenumbers

_{m}*m*= 20, 40, 60 are shown in Fig. 2. The underlying mechanism of jet formation producing these eigenfunctions can be traced to the general property that shearing of turbulence produces an upgradient momentum flux that is equivalent to an antidiffusion (Starr 1968; Farrell and Ioannou 1993b, 2007).

We can further probe the organization of turbulent eddy momentum fluxes by zonal jets by imposing an infinitesimal velocity perturbation centered at the equator of the form *δ**u**y*/Δ*y*)^{2}]. With Δ*y* = 1, the resulting eddy-turbulence-induced fractional time rate of change of the mean flow for excitation by representative zonal wavenumbers is shown in Fig. 3. Unlike the eigenfunctions shown in Fig. 2, these test functions induce momentum flux convergence that will not preserve their form. Both Figs. 2 and 3 show linear perturbation results implying that either sign of jet perturbation is equally likely to be destabilized with an *e*-folding time on the order of 300 Jovian days. It follows that either a prograde or a retrograde equatorial jet can emerge at small turbulence levels depending on initial conditions, a result consistent with the simulations of Kitamura and Ishioka (2007).

One implication of the antidiffusive upgradient momentum flux produced by perturbations to zonal velocity is that the eigenfunctions of 𝗠 are approximately the eigenfunctions of the diffusion operator, which explains the nearly harmonic form of the initial mean zonal jets that grow out of the turbulence field (cf. Fig. 2).

Turbulent eddies are distributed over all perturbation zonal scales and the eigenfunctions of the total sensitivity matrix (23) formed as the sum over contributions from all perturbation zonal wavenumbers depend on this distribution. However, as we have seen, the higher zonal wavenumbers produce disproportionately greater flux, so they dominate the tendency unless their amplitude falls off very rapidly with wavenumber.

## 4. Formation and equilibration of prograde and retrograde equatorial jets

^{3}As the excitation increases, a critical value is reached at which, with equal probability given random initial conditions, either a prograde or a retrograde jet emerges as an instability at a bifurcation point. This instability subsequently equilibrates at finite amplitude and the maximum of the equilibrium zonal jet wind speed at the equator for Jovian parameters as a function of rms velocity of the turbulence that would be maintained by the excitation if it were applied to the zero state is shown in Fig. 4. The turbulence level observed on Jupiter

^{4}(Salyk et al. 2006) is also indicated in Fig. 4. The equilibrium prograde and retrograde jet structure at selected turbulence levels along the equilibrium curves in Fig. 4 are shown in Figs. 5 and 6, together with their associated absolute and potential vorticity gradients. The jet zonal wind maximum increases with increase in turbulence, and to remain stable the jet shape adjusts to lie near the Ripa (1983) necessary condition for instability primarily by broadening the retrograde jets. The Ripa necessary condition is an inviscid criterion that requires that potential vorticity gradientchange sign for instability to arise (here

*Q*= [

*f*(

*y*) −

*D*

*u*

*h*

Zonal mean jets for Jovian parameters with a variable Coriolis parameter are shown together with their absolute and potential vorticity gradients in Fig. 7. The absolute and potential vorticity gradients are close, as they are also seen to be in observations (Read et al. 2006), and become negative predominantly where the jets are retrograde, again in agreement with observations. The potential vorticity exhibits a staircase structure as seen in Fig. 8, again in agreement with observations and simulations (Read et al. 2006; Scott and Polvani 2007). Note that negative potential vorticity gradients are not consistent with theories for jet formation based on diffusive mixing of planetary potential vorticity (Dritschel and McIntyre 2008). With these parameters the equatorial flow could equally have been retrograde, as shown in the left panel of Fig. 9. But if the excitation is increased by 15% (or equivalently the dissipation of the jet decreased by a similar factor), the retrograde jet loses structural stability and a prograde equilibrium eventually emerges, as shown in the right panel of Fig. 9. This transition is shown in the center panel of Fig. 8. The prograde jet shown at the right subsequently converges to a stable fixed point with essentially the same structure. This example shows how SSST dynamics naturally explain the phenomenon of jet coalescence and, given that SSST is linear in the eddy dynamics, demonstrates that jet coalescence is not necessarily related to a turbulent cascade.

## 5. Conclusions

Zonal jets, such as are observed on Jupiter, emerge spontaneously from turbulence in the absence of jet scale forcing. The physical mechanism giving rise to and maintaining these jets is turbulent eddy fluxes systematically organized, through interaction with the jet, to support the jet structure. This phenomenon is associated with a ubiquitous instability of turbulent fluids to jet formation, which occurs because turbulent eddies are organized by an appropriately configured perturbation zonal jet to produce exactly the momentum fluxes required to amplify that jet. This jet formation mechanism essentially emerges from the interaction between the mean state and the turbulence. The problem of capturing this emergent instability analytically is solved by SSST. The SSST equations comprise a stochastic turbulence model to obtain the eddy fluxes coupled to an evolution equation for the mean flow.

In this work we applied SSST to jet formation in the turbulent SWE on an equatorial beta plane to model emergence and equilibration of the zonal jets of Jupiter. We find for parameter values appropriate for Jupiter that these jets form spontaneously as a linear instability when turbulence reaches sufficient intensity and grow until reaching a nonlinear equilibrium. Therefore, at low turbulence levels both prograde and retrograde equatorial jets form and we find that both subsequently progress to form stable equilibria, but for sufficiently high turbulence levels and for Froude numbers used in the examples in this work only the prograde equatorial jet can be maintained as an equilibrium state. In the midlatitudes multiple jets are maintained at equilibrium with amplitude decreasing poleward. As turbulence intensity increases, the equatorial jets become increasingly asymmetric with the retrograde jet widening. This jet structure evolution with increasing turbulence intensity is characteristic of a progression that contrives to maintain the eddy stability of the jet despite increasing jet amplitude. In agreement with the observed Jovian jets (Read et al. 2006) and simulations (Scott and Polvani 2007), we find the retrograde midlatitude jets both satisfy the Ripa (1983) necessary condition for instability and exhibit pronounced change in sign of absolute vorticity gradients. We note that mixing of planetary potential vorticity cannot produce such reversals of potential vorticity gradient.

The mechanism of jet formation we have described is captured by SSST, which provides a new perspective on the interaction between turbulence and mean flows. As we have briefly discussed, the dynamics of this interaction can be understood more deeply by further analyzing the functional derivative of momentum flux divergence with mean jet velocity perturbations, a study that is in preparation.

## Acknowledgments

This work was partially supported by NSF ATM-0736022.

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^{1}

This follows from the vanishing of *q*_{y} = *β* − *u*_{yy}, implying *β* ≈ *U*/*L*^{2}, where *L* is the meridional length scale and *U* is the velocity scale.

^{2}

We have verified that the Coriolis force associated with

^{3}

The motionless state is not exactly an equilibrium because of spatial inhomogeneity of the Coriolis parameter and dissipation near the boundaries.

^{4}

This turbulence level is controlled by the stochastic forcing, parameterizing convection, and the eddy damping. These values are uncertain, but for determining the eddy fluxes the intensity of the turbulence resulting is the relevant variable, not the individual value of the excitation and damping.