## 1. Introduction

Zonal jets are prominent features of planetary, turbulent flows with well-studied examples being the banded winds of the gaseous planets (Ingersoll 1990). These large-scale flows are maintained by the momentum fluxes of the turbulent eddy field with which they coexist (Kuo 1951; Starr 1968; Vasavada and Showman 2005) and emerge spontaneously out of a background of homogeneous turbulence both in rotating-tank experiments (Read et al. 2004) and in a large number of numerical simulations of decaying (Cho and Polvani 1996) and forced turbulence (Williams 1978; Vallis and Maltrud 1993; Galperin et al. 2006).

Regarding the spontaneous formation of jets, there are several theoretical approaches discussed in the literature. These include turbulent cascades, modulational instability, mixing of potential vorticity, and statistical theories. According to the turbulent cascade approach, nonlinear eddy–eddy interactions, which are local in wavenumber space, lead to an inverse energy cascade that is “arrested” by weakly interacting Rossby waves when it reaches the Rhines scale (Rhines 1975). Because of differential rotation, the “arrest” is anisotropic in wavenumber space and allows a further upscale energy transfer to the zonal flow through a narrow region in wavenumber space (Vallis and Maltrud 1993; Nazarenko and Quinn 2009). However, observations of the atmospheric midlatitude jet (Shepherd 1987b) and numerical analysis of simulations (Nozawa and Yoden 1997; Huang and Robinson 1998) showed that the jets are maintained by spectrally nonlocal interactions rather than by a spectrally local cascade.

Modulational instability depends on eddy–eddy interactions that are nonlocal in wavenumber space: a primary meridional Rossby wave interacts with the zonal mean flow and another Rossby wave and transfers its energy directly to the zonal jet bypassing the turbulent cascade (Lorenz 1972; Gill 1974; Manfroi and Young 1999; Connaughton et al. 2010). The drawback in this approach is that it requires a constant source of finite-amplitude meridional waves. Baroclinic instability can provide such a source, but its application to almost barotropic flows (as, for example, in the Jovian atmosphere) is questionable.

Formation of jets through potential vorticity (PV) mixing envisions that Rossby wave breaking produces turbulent mixing of potential vorticity. The mixing homogenizes vorticity in localized regions, forming staircases in the vorticity gradient that correspond to mean zonal jets (Dritchel and McIntyre 2008; Dunkerton and Scott 2008). While PV staircases have been illustrated in numerical simulations and observations [see Scott and Dritchel (2012) and references therein], there are many cases in which mixing is insufficient to produce a perfect staircase structure, yet robust jets are maintained by the eddies.

Statistical equilibrium theory has also been advanced to explain emergence and formation of jets. This theory is based on the principle that turbulence tends to produce configurations that maximize entropy while conserving both energy and enstrophy. These maximum entropy configurations in two-dimensional flows assume the form of zonal jets or large-scale vortices [see review by Bouchet and Venaille (2012)]. However, the relevance of these results in planetary flows that are strongly forced and dissipated and therefore out of equilibrium remains to be shown.

Nonequilibrium statistical theories that can address such regimes are the stochastic structural stability theory (SSST; Farrell and Ioannou 2003, 2007) or the closely related second-order cumulant expansion theory (CE2) (Marston et al. 2008; Marston 2012). These theories depend on a second-order closure of the dynamics and therefore account explicitly only for the quasi-linear wave–mean flow interactions. According to SSST, an infinitesimal mean flow perturbation can organize the turbulent small-scale eddies in a way that the eddy fluxes reinforce the mean flow to produce a turbulence–mean flow cooperative instability leading to the emergence of exponentially growing jets. While the structure and the properties of the instability were studied in stochastically forced–dissipative barotropic flows (Farrell and Ioannou 2007; Bakas and Ioannou 2011; Srinivasan and Young 2012), the mechanism for the formation of jets needs to be elucidated. In this work, we undertake this task and systematically investigate the eddy–mean flow instability and its dependence on the forcing structure.

SSST has three building blocks. The first is that the eddy statistics can be obtained by retaining only the wave–mean flow interactions in the eddy dynamics. The second building block is to form, based on the quasi-linear approximation, the deterministic dynamics for the joint evolution of the eddy statistics and the mean flow. Since the eddy–eddy nonlinearity is not retained explicitly, this is equivalent to a second-order closure of the eddy cumulant expansion. The third building block is to parameterize the eddy–eddy nonlinearity as stochastic forcing and enhanced dissipation (Farrell and Ioannou 1993a; DelSole 2004). The resulting nonlinear SSST system governing the evolution of the mean flow and the eddy statistics produces bounded trajectories that are attracted to fixed points, representing steady mean flows in statistical equilibrium with their mean eddy forcing and dissipation, limit cycles, or chaotic attractors. Despite the neglect of the eddy–eddy nonlinearity, the jets in quasi-linear or SSST models were found to be in close correspondence to the jets obtained by fully nonlinear integrations in barotropic (Srinivasan and Young 2012; Constantinou et al. 2013, manuscript submitted to *J. Atmos. Sci.*), quasigeostrophic (DelSole and Farrell 1996; DelSole 1996, 2004), and primitive equations models (O'Gorman and Schneider 2007). As a result, SSST presents an accurate turbulence closure with which we can pursue theoretical study of the formation and maintenance of jets in turbulence.

Comparison of the stability analysis of the SSST system with nonlinear simulations have shown that the emergent jets can be traced to the most unstable mode of the SSST system (Srinivasan and Young 2012; Constantinou et al. 2013, manuscript submitted to *J. Atmos. Sci.*). Finite-amplitude jets can be maintained by shear straining of the turbulent field (Huang and Robinson 1998) and shear straining of the eddies by the emergent mean flow could be similarly proposed to be also responsible for the jet-forming instability. However, the shear straining mechanism was shown to produce upgradient momentum fluxes when the dissipation is weak and the eddies have time to shear over. Given that for an emerging jet the characteristic shear time scale is necessarily infinitely longer than the dissipation time scale, it needs to be shown that shear straining can produce upgradient momentum fluxes in this case as well. In addition, previous studies have shown that shearing of isotropic eddies on an infinite domain and in the absence of dissipation and *β* does not produce any net momentum fluxes (Shepherd 1985; Farrell 1987; Holloway 2010). This point was also raised by Srinivasan and Young (2012) in their study of jet formation in a barotropic *β*-plane doubly periodic channel within the framework of SSST. They have shown that isotropically forced eddies evolving in a *β*-plane constant shear flow on an infinite domain do not produce any net momentum fluxes, yet they found structural instability and jet emergence in both an infinite and a doubly periodic channel. One possibility is that finite domain effects break the symmetry of isotropy and can lead to upgradient fluxes (Shepherd 1987a; Cummins and Holloway 2010). However, since the results in the infinite domain and the periodic channel agree in Srinivasan and Young (2012) another mechanism should be responsible for producing the upgradient fluxes in these simulations.

In this work we identify physical mechanisms that promote or obstruct jet formation. We show that shear straining of small-scale eddies by the local shear of an infinitesimal sinusoidal mean flow, as described by Orr dynamics in a *β* plane, intensifies in general the jet. We show that a mean flow velocity perturbation interacting with an anisotropic eddy field induces momentum fluxes that reinforce the mean flow, exactly as if the mean flow were acted by a negative viscosity. We also show that a mean flow velocity perturbation interacting with an isotropic eddy field induces upgradient momentum fluxes caused by changes in the propagation of the eddies that act as a negative hyperviscosity on the mean flow.

## 2. Statistical wave–mean flow barotropic dynamics

*β*plane. Relative vorticity

*q*(

*x*,

*y*,

*t*) evolves according to

*J*(

*A*,

*B*) =

*A*−

_{x}B_{y}*A*,

_{y}B_{x}*ψ*is the streamfunction, and

*r*is the coefficient of linear dissipation that typically parameterizes Ekman drag. The forcing term

*f*arises from processes that are missing from the barotropic dynamics (e.g., cascade of energy from baroclinic to barotropic eddies, or small-scale convection) and is typically taken as a spatially homogeneous, random stirring. We decompose the fields into their zonal mean component, denoted with capital letters, and perturbations from this mean, denoted with primes. Under this decomposition and assuming a vanishing external excitation for the zonal mean flow, (1) is split into two equations governing the evolution of the perturbations

_{e}*q*′ and the zonal component of the zonal mean velocity

*U*:

*u*′,

*υ*′) = (−∂

_{y}

*ψ*′, ∂

_{x}

*ψ*′) denote the zonal and meridional eddy velocities, respectively. The overbar denotes a zonal average and

*f*

_{nl}can be either neglected (Marston 2012) or, in order to parameterize nonlinear cascading processes, it can be represented as a random broadband forcing augmented with an additional effective eddy damping to conserve energy (DelSole 2001; Farrell and Ioannou 2009). In this work, both forcing terms

*f*=

*f*+

_{e}*f*

_{nl}will be represented as a stochastic excitation without distinction.

*Q*is a function of the differences

*a*=

_{i}*a*(

_{i}**x**

_{i},

*t*), with

*i*= 1, 2 to refer to the value of the variable

*a*at the points

**x**

_{i}= (

*x*,

_{i}*y*). To calculate the equation for the evolution of the vorticity covariance function

_{i}**x**

_{i}and

*ψ*= Δ

^{−1}

*q*and consequently the velocities are

*u*= −∂

_{y}Δ

^{−1}

*q*and

*υ*= ∂

_{x}Δ

^{−1}

*q*. Multiplying (5) for

*A*

_{1}and

*A*

_{2}commute, and that

*C*is a function of

*x*

_{1}−

*x*

_{2}, since

*A*

_{1},

*A*

_{2}, and

*Q*are all homogeneous in

*x*. Note also that for delta correlated forcing, the ensemble average enstrophy injection rate,

**x**

_{1}=

**x**

_{2}means that the expression in parenthesis, which is a function of the two points

**x**

_{1}and

**x**

_{2}, is calculated at the same point. We make the ergodic assumption that the ensemble average is equal to the zonal average; that is, we assume that

*C*and

*U*. This coupled system constitutes a second-order closure for the dynamics and is the basis of the SSST.

*x*the Laplacian takes the form

*C*by

*U*and covariance

^{E}*C*. For a spatially homogeneous forcing

^{E}*Q*we always have the equilibrium

*β*-plane turbulence (Vallis and Maltrud 1993; Read et al. 2007; Scott and Polvani 2008). In the context of SSST, this phenomenon can be addressed by performing stability analysis of the equilibrium

*U*,

^{E}*C*using the SSST equations. A small perturbation mean flow

^{E}*δU*and perturbation covariances

*δC*and

*δ*Ψ about this equilibrium obey the linear equations

*U*and

^{E}*C*is consequently reduced to the eigenanalysis of the linearized equations in (15) and (16). Eigenanalysis of (15) and (16) reveals that the homogeneous equilibrium is unstable when the forcing amplitude exceeds a threshold that depends on the damping and the forcing structure. A jet-emerging instability occurs if a seed mean flow organizes the eddies so that the eddy fluxes reinforce it, producing a positive feedback that results in the exponential growth of the jet. This eddy–mean flow feedback process is therefore crucial for the instability, and will be studied in this work in detail.

^{E}## 3. Response of the eddy fluxes to mean flow perturbations

*δU*in order to illuminate the nature of the structural instability leading to jet formation. The perturbation in vorticity covariance

*δC*that is induced by

*δU*can be estimated immediately by assuming that the system (15) and (16) is very close to the stability boundary, so that the growth rate is small. We choose this adiabatic limit because it was shown in Bakas and Ioannou (2011) that a necessary condition for structural instability in the case of jet formation is the existence of upgradient momentum and vorticity fluxes in this limit. In this case the mean flow evolves slowly enough that it remains in equilibrium with the eddy covariance. If the marginally unstable state has eigenvalues with zero imaginary part, then

*dδC*/

*dt*≃ 0, and the streamfunction perturbation covariance function

*δ*Ψ obtained from (15) in this limit is

*δ*Ψ into two parts,

*δ*Ψ

^{ad}and

*δ*Ψ

^{cu}, is instructive because it isolates two physical processes that contribute to the perturbation covariance: advection of the eddy vorticity (the equilibrium vorticity covariance) by the mean flow perturbation and advection of the perturbed mean flow vorticity −

*δU*′ by the eddies. We can thus calculate distinct momentum fluxes originating from these two processes, yielding the total perturbation momentum flux

*δU*= sin(

_{n}*ny*), which are indexed by the meridional wavenumber of the mean flow

*n*(which is a continuous variable for the infinite domain). To calculate the momentum fluxes that result from (17) for a mean flow perturbation of the form

*δU*= sin(

*ny*) that explicitly and clearly illustrates the behavior of the eddy fluxes, we consider the limit in which the scale of the mean flow 1/

*n*is much larger than the scale of the forcing

*L*so that

_{f}*βL*is at most of the same order as the dissipation time scale 1/

_{f}*r*, so that

^{1}In this limit the momentum fluxes are approximately given by

*A*,

_{S}*A*,

_{β}*A*, and

_{C}*A*are coefficients that depend on the spectral characteristics of the forcing and are given by (B11) and (B12) (cf. appendix B). The same result is also obtained if we assume that the eddies evolve in a slowly varying flow according to the local shear and according to the local mean vorticity gradient (cf. appendix B). This shows that in the limit of

_{P}*A*,

_{S}*A*,

_{β}*A*, and

_{C}*A*for two cases of forcing. We first treat the case of the isotropic forcing considered by Srinivasan and Young (2012):

_{P}*J*

_{0}is the zeroth-order Bessel function. The representation of this isotropic forcing in wavenumber space is a delta function ring of radius

*K*, for which

_{f}*L*= 1/

_{f}*K*. This isotropic forcing has been typically used in studies of barotropic turbulence and is thought to roughly represent convective forcing at scale

_{f}*L*(Scott and Polvani 2008). For this forcing, all coefficients

_{f}*A*,

_{S}*A*, and

_{C}*A*are exactly zero and the only nonzero contribution to the momentum fluxes results from

_{P}*A*(cf. appendix B). The leading-order contributions are

_{β}*δU*resulting in a hyperviscous momentum flux divergence that tends to reinforce the mean flow and is therefore destabilizing:

*σ*of the energy density in a constant flow of unit velocity. In this case, all the coefficients

*A*,

_{S}*A*,

_{C}*A*, and

_{P}*A*are nonzero and the momentum fluxes for a zonally confined forcing (

_{β}*k*≫ 1) are to leading order given by

_{f}δ^{2}

*β*. This implies that there is a tendency to form jets also in the absence of

*β*as seen, for example, in numerical simulations (Kramer et al. 2008; Bouchet and Simonnet 2009). The other terms in (25) (including the one that is destabilizing for isotropic forcing) act as hyperviscosity and oppose jet formation but are subdominant.

*R*, above topography of small elevation

_{d}*η*, and consider for simplicity topography consisting of zonal ridges (i.e., take

*η*to be zonally invariant). Then (28) implies that the vorticity fluxes due to advection of the mean potential vorticity

## 4. Analysis of the dynamics underlying the eddy fluxes

*dδC*/

*dt*≃ 0. The perturbation momentum fluxes induced by a sinusoidal mean flow perturbation

*δU*= sin(

*ny*) can be alternatively calculated from

*t*produced by the initial perturbation

*ξ*and

*y*direction. We will now show that

*G*takes the form of a wavepacket for both the isotropic and anisotropic forcing studied in the previous section.

*x*) = 0 when

*x*< 0 and Θ(

*x*) = 1 when

*x*≥ 0]. When the width of the ring Δ

*K*around wavenumber

*K*goes to zero, this forcing approaches the narrow band ring forcing in (21) treated in section 3. For this forcing,

_{f}*G*has the wavepacket form (cf. appendix C)

*G*(

*k*,

*y*−

*ξ*) consists of a carrier wave with wavenumbers (

*k*,

*l*

_{0}) and amplitude

*B*(

*k*), which is modulated in the

*y*direction by the wavepacket envelope

*h*(

*y*). Consider now the anisotropic forcing in (24). It is straightforward to show that

*G*assumes the same form as (33) but with (cf. appendix C)

*nL*≪ 1 and

_{f}*δU*, the eddies are also dissipated before they shear over. As a result, the waves evolve to a good approximation according to the local dynamics (cf. appendix B). That is, the vorticity of the eddy that is initially localized around

*ξ*is advected by the local velocity

*nL*≪ 1 does not hold, the eddies will be affected by the mean shear and mean

_{f}*Q*within their extend. If the propagation time scale is large compared to the dissipation time scale, so that

_{y}*Q*within the extent of their propagation.

_{y}In section 3, we found that the processes of eddy vorticity advection and advection of the mean flow vorticity by the eddies can be separated, and that their contribution to the total momentum fluxes, denoted as *β* plane. Similarly, for *Q _{y}* and calculate the momentum fluxes due to advection of the vorticity gradient of the mean flow by the eddies from the evolution of the wavepacket in a fluid with no mean flow but with a vorticity gradient

*Q*given by (37). The ensemble mean momentum fluxes will then be given by (29).

_{y}### a. Shear wave dynamics

*δU*=

*α*(

*y*−

*ξ*), where

*α*= (

*dδU*/

*dy*)

_{ξ}is the local shear at each latitude

*ξ*of the emerging flow. The momentum flux of each wavepacket is

*l*=

_{t}*l*

_{0}−

*αkt*, decreases with time as the wave is sheared over, while the group velocity of the wavepacket on the

*β*plane,

*c*= 2

_{g}*βA*, is proportional to

_{M}*A*. The position of the packet is given by

_{M}*t*=

*l*

_{0}/

*kα*, and subsequently propagates southward and asymptotically reaches its critical layer where it surrenders its momentum to the mean flow. On the other hand, a wavepacket with phase lines tilted with the shear propagates southward toward its critical layer while it continuously loses its momentum to the mean flow (Boyd 1983; Tung 1983).

In the presence of strong shear and weak damping, the shear time scale is much smaller than the dissipation time scale and the eddies quickly shear over into the decaying phase that lasts longer than the growing phase. As a result, they decay on average and the mean flow is accelerated by the upgradient momentum fluxes of the shear waves. This is the mechanism for maintaining finite-amplitude jets by an eddy field that is typically referred to as the Orr mechanism (Farrell and Ioannou 1993b; Huang and Robinson 1998). This process ceases to produce upgradient fluxes when the eddy field is isotropic (Shepherd 1985; Farrell 1987).

*r*≪ 1/

*α*. In the limit of a weak mean vorticity gradient that we also consider (

*α*/

*r*≪

*βL*/

_{f}*r*≪ 1), the dominant contribution to the time integral in (30) comes from short times, since the perturbation is rapidly attenuated by friction before it propagates or it is sheared over. As a result, the average momentum flux distribution will be determined by two factors: the small change in the amplitude of the fluxes

*A*due to shearing over a dissipation time scale and the small change in the position of the packet that occurs during the same period. For short times (

_{M}*αt*≪

*rt*) the variation in the momentum flux amplitude and the location of the wavepacket due to shearing is

*p*=

*y*−

*ξ*. The first term,

*ξ*. The second and third terms arise because of shearing of the wavepacket and correspond to the contribution of the change in the amplitude of the fluxes

We can qualitatively assess the changes in the distribution of the fluxes obtained in (41) by examining how the amplitude of the fluxes and the group velocity of the packets change as the phase lines of the carrier wave are sheared over. Let *θ _{t}* = arctan(

*l*/

_{t}*k*) be the angle at time

*t*of the phase lines of the carrier wave of the packet with the

*y*axis. Figures 2a and 3a illustrate the amplitude of the momentum fluxes

*A*(Fig. 2a) and the group-velocity

_{M}*c*(Fig. 3a) at the instance of time at which the phase line orientation is

_{g}*θ*. We first study the effect of the amplitude change by ignoring propagation. Consider a wavepacket starting at some point along the

_{t}*θ*axis with initial angle

_{t}*θ*

_{0}= arctan(

*l*

_{0}/

*k*). Then the filled circle shown in Fig. 2a gives the initial value of the momentum fluxes. As the packet is sheared over with time,

*θ*decreases monotonically (owing to the monotonically decreasing

_{t}*l*) and the fluxes at a later time are given by the open circle. Since the wave packet is rapidly dissipated, the integrated momentum fluxes over its lifetime will be given to a good approximation by the change in the fluxes occurring over the dissipation time scale 1/

_{t}*r*that is incremental in shear time units. The change in the fluxes will thus be proportional to the local derivative of the curve in Fig. 2a. As a result, the momentum flux of a wavepacket with |

*θ*

_{0}| <

*π*/6 (corresponding to

^{3}This is also illustrated in Fig. 2b, showing how the momentum flux of a wavepacket with a Gaussian distribution of vorticity with latitude changes as the wave shears over if we ignore propagation. Compared to an unsheared wavepacket, this process leads to the upgradient momentum flux surplus shown in Fig. 2c. The opposite occurs for waves excited in regions I and IV (with |

*θ*

_{0}| <

*π*/6 corresponding to

(a) Momentum fluxes *A _{M}*(

*t*) of wavepackets in a constant shear flow as a function of the angle

*θ*= arctan(

_{t}*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. The wavenumber is given by

*l*=

_{t}*l*

_{0}−

*αkt*, where (

*k*,

*l*

_{0}) is the initial central wavenumber of the wavepacket and

*α*is the shear. A wavepacket starting with an inclination at a certain angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its fluxes at a later time will be given by the open circle. The vertical lines separate the regions with |

*θ*| <

_{t}*π*/6 (II and III) and |

*θ*| >

_{t}*π*/6 (I and IV). At

*θ*= ±

_{t}*π*/6, the momentum flux has peak magnitude for wavepackets excited with equal vorticity. The central wavenumber of the packet is

*t*= 0.2/

*r*is shown. The wavepacket has initial vorticity

*K*= 1, |

_{f}*θ*

_{0}| <

*π*/10, and |

*B*| = 1. The shear and dissipation time scales are taken as equal (

*α*=

*r*= 0.1) for illustration purposes. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of the amplitude change

*α*= 10

^{−3}, while the rest of the parameters are as in (b).

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

(a) Momentum fluxes *A _{M}*(

*t*) of wavepackets in a constant shear flow as a function of the angle

*θ*= arctan(

_{t}*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. The wavenumber is given by

*l*=

_{t}*l*

_{0}−

*αkt*, where (

*k*,

*l*

_{0}) is the initial central wavenumber of the wavepacket and

*α*is the shear. A wavepacket starting with an inclination at a certain angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its fluxes at a later time will be given by the open circle. The vertical lines separate the regions with |

*θ*| <

_{t}*π*/6 (II and III) and |

*θ*| >

_{t}*π*/6 (I and IV). At

*θ*= ±

_{t}*π*/6, the momentum flux has peak magnitude for wavepackets excited with equal vorticity. The central wavenumber of the packet is

*t*= 0.2/

*r*is shown. The wavepacket has initial vorticity

*K*= 1, |

_{f}*θ*

_{0}| <

*π*/10, and |

*B*| = 1. The shear and dissipation time scales are taken as equal (

*α*=

*r*= 0.1) for illustration purposes. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of the amplitude change

*α*= 10

^{−3}, while the rest of the parameters are as in (b).

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

(a) Momentum fluxes *A _{M}*(

*t*) of wavepackets in a constant shear flow as a function of the angle

*θ*= arctan(

_{t}*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. The wavenumber is given by

*l*=

_{t}*l*

_{0}−

*αkt*, where (

*k*,

*l*

_{0}) is the initial central wavenumber of the wavepacket and

*α*is the shear. A wavepacket starting with an inclination at a certain angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its fluxes at a later time will be given by the open circle. The vertical lines separate the regions with |

*θ*| <

_{t}*π*/6 (II and III) and |

*θ*| >

_{t}*π*/6 (I and IV). At

*θ*= ±

_{t}*π*/6, the momentum flux has peak magnitude for wavepackets excited with equal vorticity. The central wavenumber of the packet is

*t*= 0.2/

*r*is shown. The wavepacket has initial vorticity

*K*= 1, |

_{f}*θ*

_{0}| <

*π*/10, and |

*B*| = 1. The shear and dissipation time scales are taken as equal (

*α*=

*r*= 0.1) for illustration purposes. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of the amplitude change

*α*= 10

^{−3}, while the rest of the parameters are as in (b).

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

(a) The group velocity of the wavepackets in a constant shear flow as a function of the angle *θ _{t}* = arctan(

*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. A wavepacket starting at an angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its group velocity at a later time will be given by the open circle. The regions I–IV are as in Fig. 2a,

*β*= 0.6 for illustration purposes, and

*K*= 1. (b) Comparison of the momentum fluxes of an unsheared wavepacket excited in regions II (thick solid line) and III (thin solid line) to the momentum fluxes of a sheared wavepacket shown by the corresponding dashed lines, when only the change in propagation is taken into account. A snapshot of the fluxes at

_{f}*t*= 0.2/

*r*is shown and the rest of the parameters are as in Fig. 2b. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of propagation is taken into account. The planetary vorticity gradient is

*β*= 0.1 and the rest of the parameters are as in Fig. 2c.

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

(a) The group velocity of the wavepackets in a constant shear flow as a function of the angle *θ _{t}* = arctan(

*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. A wavepacket starting at an angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its group velocity at a later time will be given by the open circle. The regions I–IV are as in Fig. 2a,

*β*= 0.6 for illustration purposes, and

*K*= 1. (b) Comparison of the momentum fluxes of an unsheared wavepacket excited in regions II (thick solid line) and III (thin solid line) to the momentum fluxes of a sheared wavepacket shown by the corresponding dashed lines, when only the change in propagation is taken into account. A snapshot of the fluxes at

_{f}*t*= 0.2/

*r*is shown and the rest of the parameters are as in Fig. 2b. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of propagation is taken into account. The planetary vorticity gradient is

*β*= 0.1 and the rest of the parameters are as in Fig. 2c.

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

(a) The group velocity of the wavepackets in a constant shear flow as a function of the angle *θ _{t}* = arctan(

*l*/

_{t}*k*) between the phase lines of the central wave and the

*y*axis. A wavepacket starting at an angle

*θ*

_{0}(filled circle) will transverse this graph toward the left and its group velocity at a later time will be given by the open circle. The regions I–IV are as in Fig. 2a,

*β*= 0.6 for illustration purposes, and

*K*= 1. (b) Comparison of the momentum fluxes of an unsheared wavepacket excited in regions II (thick solid line) and III (thin solid line) to the momentum fluxes of a sheared wavepacket shown by the corresponding dashed lines, when only the change in propagation is taken into account. A snapshot of the fluxes at

_{f}*t*= 0.2/

*r*is shown and the rest of the parameters are as in Fig. 2b. (c) The difference in momentum fluxes between a sheared and an unsheared wavepacket calculated over their life cycle, when only the effect of propagation is taken into account. The planetary vorticity gradient is

*β*= 0.1 and the rest of the parameters are as in Fig. 2c.

Citation: Journal of the Atmospheric Sciences 70, 7; 10.1175/JAS-D-12-0102.1

We now consider the effect of propagation on the momentum fluxes while ignoring the change in the amplitude. A wavepacket starting in region III will propagate toward the north, as shown by the filled circle in Fig. 3a. Because shearing induces a decrease in the magnitude of the group velocity (open circle in Fig. 3a) the wavepacket will flux its momentum from southern latitudes compared to when it moved in the absence of the shear flow. This is shown in Fig. 3b, illustrating the distribution of momentum fluxes of an unsheared and a sheared perturbation whose amplitudes are constant. Figure 3c plots this difference,

*ξ*and all zonal wavenumbers we obtain

*nL*≪ 1), the shear

_{f}*dδU*/

*dξ*over the wavepacket envelope

*ξ*=

*y*. The third term in (42) is

*A*term in (19). The net fluxes are shown to be exactly zero for isotropic forcing because the gain in momentum occurring for |

_{S}*θ*

_{0}| <

*π*/6 (waves excited in regions II and III) is fully compensated by the loss in momentum for |

*θ*

_{0}| >

*π*/6 (waves excited in regions I and IV). Full compensation occurs because we are equally exciting all possible wave orientations. This exact cancelation is a peculiarity of isotropic forcing in an unbounded domain. A finite domain, as is the case in physically realizable flows, can affect this symmetry and lead to partial cancelation and to upgradient fluxes, as was noted in previous studies (Shepherd 1987a; Cummins and Holloway 2010). In (46), the net momentum fluxes are produced by the

*θ*

_{0}| <

*π*/6 is overcompensated by the gain in momentum for |

*θ*

_{0}| >

*π*/6. The momentum fluxes are hyperdiffusive and the wavepacket analysis reproduces (22) modulo a factor of 2.

*l*

_{0}= 0 in wavenumber space). We have therefore revealed the dynamics underlying the first two terms

^{4}in (19). In summary, the change in the amplitude of the momentum fluxes caused by shearing of the eddies leads to upgradient antidiffusive ensemble mean momentum fluxes that reinforce the mean flow except for isotropic forcing in which case it has no effect. The change in the group velocity of the eddies due to shearing leads to hyperdiffusive fluxes with a positive or negative coefficient depending on the characteristics of the forcing.

### b. Dynamics of wave propagation in the presence of a mean vorticity gradient

*Q*=

_{y}*β*−

*γ*(

*y*−

*ξ*), with

*d*

^{2}

*δU*/

*dy*

^{2}|

_{ξ}= 0 and

*d*

^{3}

*δU*/

*dy*

^{3}|

_{ξ}=

*γ*. Under the slowly varying approximation, the wavepacket has a local phase speed

*Q*, resulting in a decreasing wavenumber

_{y}*l*and

_{t}*η*satisfy (D2) in this case. For small times,

*l*satisfies the heuristically derived (48) and the dynamics are homomorphic to the shear wave dynamics described in section 4a.

_{t}*β*, arises solely because of the change in the amplitude of the momentum fluxes. This result can be comprehended qualitatively by examining again Fig. 2a, showing how the momentum fluxes change with

*θ*. The only difference in this case is the time that it takes for a wave to transverse this graph toward the left, as

_{t}*l*decreases monotonically but by a different amount compared to the case in section 4a. However, this difference is not relevant with regard to the momentum flux changes, as we are interested in changes within the dissipation time scale that are infinitesimal compared to the evolution time scale for

_{t}*l*[which is

_{t}*O*(1/

*α*) ≫ 1 in section 4a and

*O*(1/

*γ*) ≫ 1 here]. As a result, the momentum fluxes of a wavepacket excited in regions II and III will again decrease within the dissipation time scale. This leads to the same flux surplus shown in Fig. 2c when

*γ*=

*α*. However, the amplitude is proportional to the third derivative of the mean flow

*γ*, rather than proportional to the shear, and the momentum fluxes are therefore downgradient. The opposite occurs for waves excited in regions I and IV, producing upgradient fluxes.

*θ*

_{0}| <

*π*/6 is fully compensated by the gain in momentum for |

*θ*

_{0}| >

*π*/6. For the anisotropic forcing in (24), we obtain

*Q*produces hyperdiffusive fluxes with an amplitude independent of

_{y}*β*and therefore corresponds to the

*A*term in (20).

_{P}## 5. Conclusions

Large-scale zonal jets are commonly observed to spontaneously emerge in turbulent fluids. The mechanism for jet formation in a barotropic *β* plane under homogeneous stochastic forcing was examined in this work within a statistical wave–mean flow interaction framework. In this framework, the eddy–eddy nonlinearity is parameterized or ignored. This approximation leads to a deterministic system for the coevolution of the zonal mean jet and the ensemble mean covariance of the perturbation fields. This dynamics is the subject of stochastic structural stability theory (SSST) (Farrell and Ioannou 2003).

We derived in this work the SSST system with the continuous formulation of Srinivasan and Young (2012) and derived the correspondence with the matrix formulation of the same equations. We then discussed the structural stability of a homogeneous equilibrium maintained against dissipation by a spatially homogeneous and delta-correlated-in-time stochastic excitation. It is known that in such flows on a *β* plane, the homogeneous state is structurally unstable when a critical value of forcing is exceeded and zonal jets emerge. We focused our analysis close to this bifurcation point in order to identify the processes that lead to the emergence of jets. Investigation of the ensemble mean momentum fluxes revealed that the eddy–mean flow dynamics can be split into two distinct processes: advection of the eddy vorticity by the weak mean flow and advection of the vorticity of the mean flow by the eddies. Eddy vorticity advection was found to lead to hyperdiffusive fluxes with a negative diffusion coefficient when the stochastic forcing is isotropic and to antidiffusive fluxes when the forcing is anisotropic. In both cases this leads to the enhancement of the mean flow and to instability. On the other hand, advection of the mean vorticity by the eddies was found to have no effect to leading order when the forcing is isotropic and to lead to hyperdiffusive fluxes hindering jet formation when the forcing is anisotropic.

These processes were then examined in detail by studying the momentum fluxes induced by an ensemble of wavepackets in the presence of an infinitesimal sinusoidal mean flow. Assuming slow mean flow variations, we estimated the contribution of shearing of the wavepackets by the local shear and the contribution of wavepacket propagation under the inhomogeneous vorticity gradient in the ensemble mean momentum fluxes for both isotropic and anisotropic forcing. These calculations were performed in the physically relevant limit for the emergence of jets of small dissipation time scale compared to the shear time scale.

Shearing of the eddies in the manner described by the Orr dynamics in a *β* plane was found to have two effects. The first effect is that it changes the amplitude of the fluxes in accordance with conservation of vorticity. This process leads to upgradient fluxes with an amplitude proportional to the shear unless the forcing is isotropic, in which case it produces no fluxes at all. This process underlies the negative viscosity characteristic of the fluxes in the case of anisotropic forcing. The second effect is that it changes the group velocity of the wavepacket compared to an unsheared perturbation propagating in a *β* plane. This process leads to momentum fluxes with an amplitude proportional to the third derivative of the mean flow that are upgradient for an isotropic forcing and downgradient for an anisotropic forcing. As a result, this process underlies the negative hyperviscosity characteristic of the fluxes in the case of isotropic forcing. In any case, the driving mechanism for the emergence of jets was found to be shearing of the eddies by the local shear in a *β* plane.

On the other hand, refraction of the eddies in the manner described by ray tracing of Rossby waves propagating under the inhomogeneous local mean vorticity gradient was found to change the amplitude of the fluxes according to wave action conservation. This process produces downgradient fluxes with an amplitude proportional to the third derivative of the mean flow, unless the forcing is isotropic, in which case it has no effect. As a result, this process underlies the hyperdiffusive action of the fluxes in the case of anisotropic forcing. Nevertheless, this process can be jet forming in the presence of topography with zonal mean flows emerging in regions where topography enhances *β*.

## Acknowledgments

This research was supported by the EU FP-7 under the PIRG03-GA-2008-230958 Marie Curie Grant. The authors acknowledge the hospitality of the Aspen Center for Physics supported by the NSF (under Grant 1066293), where part of this work was written. The authors would also like to thank Kaushik Srinivasan and Bill Young for fruitful discussions and an early version of their manuscript and Brian Farrell and Freddy Bouchet for stimulating discussions on the results of this work.

## APPENDIX A

### Discrete Formulation of the SSST Equations

*y*evolve according to

_{j}*f*(

*Ã*

_{jk}denotes the

*k*wavenumber operator

*A*acting at points

*y*). A real vorticity field requires that

_{j}*k*= 0 and

*k*′ = 0 excluded in the summation. Since

*C*is a function of

*x*

_{1}−

*x*

_{2}, only terms with

*k*′ = −

*k*contribute in (A3) and

*k*= 0 excluded. A similar decomposition is obtained for the forcing covariance, with the definition

**q**

_{k}and row

_{k}and

_{y}and

*y*

_{1}=

*y*

_{2}. This matrix system is equivalent to (12) and (13).

## APPENDIX B

### Calculation of Momentum Fluxes for a Sinusoidal Flow

*δU*= sin(

_{i}*ny*). The perturbation streamfunction covariance in the adiabatic limit is determined from

_{i}*C*=

^{E}*Q*/2

*r*, and

*δ*Ψ =

*δ*Ψ

^{ad}+

*δ*Ψ

^{cu}. The first term

*δ*Ψ

^{ad}is the contribution to the perturbation covariance from the advection of the equilibrium vorticity covariance by the perturbation mean flow. The second term,

*δ*Ψ

^{cu}, is the contribution from the advection of the vorticity of the perturbed flow by the eddies at equilibrium. Since this is a linear equation for the perturbation velocity, we choose the mean flow perturbation

*δ*Ψ

^{ad}and

*δ*Ψ

^{cu},

*Q*. Introducing in (B4) the expression for

*δ*Ψ

^{ad}and

*δ*Ψ

^{cu}to the momentum fluxes. Because the covariances satisfy the exchange symmetry

*Q*(

*x*

_{1},

*x*

_{2},

*y*

_{1},

*y*

_{2}) =

*Q*(

*x*

_{2},

*x*

_{1},

*y*

_{2},

*y*

_{1}), which is the equivalent expression of the hermiticity of the corresponding covariances in the matrix formulation, we obtain

*Q*is a real function and

*k*and

*l*in (B6) and using (B7), we obtain that Λ

_{+}= −Λ

_{−}. We assume that the smallest scale in which the forcing has significant power is

_{+}= −Λ

_{−}and shifting the origin of the

*l*axis

*l*→

*l*−

*n*/2 in the integral reduces (B5) to

*δ*Ψ

^{ad}and

*δ*Ψ

^{cu}to the momentum fluxes, so that

*Q*is a separable function of

*βL*is at most of the same order as the dissipation time scale 1/

_{f}*r*(

*βL*/

_{f}*r*< 1), so that

*F*is

*rK*

^{6}, while all the other terms are order

^{B1}Expanding

*F*in powers of

*ñ*might appear in the parenthesis in (B10) (K. Srinivasan 2013, personal communication). We will now show that the approximate expression in (B10) can also be obtained by considering a slowly varying mean flow

*U*=

_{i}*U*(

_{i}*νy*), where

_{i}*ν*≪ 1, for which

*y*. They also advect the local mean vorticity that has to leading order a linear gradient with respect to latitude and is also slowly varying. A two-scale perturbation expansion of (B1) with slow variable

*Y*=

*νy*and

*δ*Ψ =

*ν*Ψ

^{1}+

*ν*

^{2}Ψ

^{2}+

*ν*

^{3}Ψ

^{3}+ … gives that the Fourier amplitudes of Ψ

^{i}to the third order are

^{B2}

*U*(

*νy*) = sin(

*ny*). This proves that the limit

*A*,

_{S}*A*,

_{β}*A*, and

_{C}*A*for the two cases of stochastic forcing discussed in section 3. The first is the isotropic ring forcing in (21) with

_{P}*L*= 1/

_{f}*K*and the second is the anisotropic forcing in (24) with

_{f}*L*= min(1/

_{f}*k*,

_{f}*δ*). The amplitude

*σ*of the energy in a constant flow of unit velocity. To calculate

_{f}:

*σ*/2 according to the normalization yielding

*A*=

_{S}*A*=

_{P}*A*= 0 and

_{C}*k*≫ 1, (B20)–(B23) are approximately equal to

_{f}δ*k*≪ 1, (B20)–(B23) are approximately equal to

_{f}δ*δ*→ 0, the only nonzero coefficient is

## APPENDIX C

### Eddy Fluxes Produced by an Ensemble of Wave Packets

*δU*around the statistical equilibrium in (14) and calculate the perturbation covariance

*δC*that is induced by

*δU*. The solution of (8) for stationary mean flows, and time invariant

*A*, is

_{i}*t*→ ∞ limit of (C1). In the adiabatic limit, in which the covariances at all times assume their equilibrium value (

*dδC*/

*dt*≃ 0),

*δC*is the difference between covariance

*C*that results at steady state when the mean flow is

^{δU}*δU*and the equilibrium covariance

*C*. That is, we can write

^{E}*δU*= sin(

_{i}*ny*) is the mean flow perturbation. Substituting (C2) into (9), we obtain the alternative expression for the momentum fluxes

_{i}*f*and obtain the forcing covariance as

*Q*is homogeneous and depends only on

*x*

_{1}−

*x*

_{2}and

*y*

_{1}−

*y*

_{2}, the ensemble mean Fourier amplitudes satisfy

*Q*implies that

*t*produced by the initial monochromatic perturbation

*G*(

*k*,

*y*)

*e*, which is localized at latitude

^{ikx}*ξ*. We have therefore shown that the perturbation momentum fluxes can be calculated as follows: take an ensemble of perturbations

*G*(

*k*,

*y*)

*e*, each localized around different latitudes, calculate the momentum fluxes over their life cycles, and then add their relative contribution, as if the waves evolved independently, to obtain the ensemble mean perturbation fluxes. We will now show that for both the ring forcing in (21) and the anisotropic forcing in (24), the perturbations

^{ikx}*G*can be interpreted as wavepackets.

*K*in wavenumber space. The ring sector |

*K*−

*K*| ≤ Δ

_{f}*K*is equivalently determined by the inequalities

*k*. To calculate the momentum fluxes in closed form, we consider the modification of (32):

*δ*(

*k*). For the forcing in (C14) in the limit Δ

*K*≪ 1, we integrate (C9) to obtain to a good approximation that

*G*is a wavepacket with central wavenumber

*l*

_{0}and −

*l*

_{0}in the momentum fluxes, while the last two terms are the interference terms between these two waves. By taking the limit Δ

*K*→ 0,

*δ*→ 0 the interference terms go to zero, resulting in zero net momentum fluxes imposed at

*t*= 0. For the same reason, the interference terms make no contribution when the perturbations evolve in a shear flow. So we can take

*G*to be given by (33), for which we consider only the single wave

*l*

_{0}. The relative contribution of −

*l*

_{0}in the ensemble mean momentum fluxes is taken into account by adding the corresponding term in (41) for −

*l*

_{0}in order to obtain (42). Consider now the anisotropic forcing in (24), for which

*G*is given by

## APPENDIX D

### Wave Propagation in an Inhomogeneous Medium

*Q*=

_{y}*β*−

*γ*(

*y*−

*ξ*), with

*y*and the slowly varying wavenumber

*l*(

_{t}*t*) along a ray are given by the standard ray-tracing equations (Andrews et al. 1987)

*d*/

_{g}*dt*denotes time differentiation along a group-velocity ray. We obtain from (D1) that the wavenumber

*l*and the displacement of the wavepacket

_{t}*η*(

*t*) ≡

*y*satisfy

*l*

_{0}=

*l*(0),

_{t}*ξ*the initial position of the wavepacket, and

*y*component of the group velocity. The wavepacket therefore evolves as

*B*(

_{t}*t*) is the time-dependent amplitude with

*B*(0) =

_{t}*B*. To leading order, the spatial density of wave action along rays satisfies the equation

*E*(

*t*) is the energy density of the wavepacket following a ray that is given by

*B*=

_{t}*B*(0)

*e*

^{−rt}. As a result, the momentum fluxes produced by the wavepacket are given by (38). The dominant contribution to (30) comes from small times in the limit

*k*and

*l*

_{0}, we obtain (49).

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^{1}

In Earth's atmosphere, the size of the eddies and the jet are about 10^{3} km and 4 × 10^{3} km, respectively, and the eddy dissipation time scale is about 2 days, so that

^{2}

Similar results are also obtained in the opposite limit of a meridionally confined forcing (*k _{f}δ* ≪ 1). The only exception is the limiting case of an uncorrelated forcing (

*δ*→ 0), for which the only nonzero coefficient is

*A*leading to downgradient hyperdiffusive fluxes (cf. appendix B).

_{P}^{3}

Note that the angles *θ _{m}* = ±

*π*/6 correspond to the orientations at which a wave with initial vorticity

*B*(

*k*) maximizes the momentum flux amplitude. If the wave were introduced with initial energy

*E*(

*k*), then the momentum fluxes of the carrier wave ignoring dissipation would be given by

*θ*= ±

_{m}*π*/4.

^{4}

To investigate the third term, we need to take into account the third derivative of *δU* in (36). However, since this turns out to be a stabilizing term for the cases considered, we will not pursue this further.

^{B1}

The limit *βL _{f}*/

*r*< 1 ensures that the first terms in the denominator are at least order

*rK*

^{6}.