,Farrell-87,Holloway-10} and should have no effect 
on the S3T instability \citep{Srinivasan-Young-12}. Therefore another mechanism should be responsible 
for producing the upgradient fluxes in the case of an isotropic forcing.



In order to investigate the eddy-mean flow dynamics underlying the S3T instability, we calculate the
vorticity flux divergence that is induced when the statistical equilibrium (\ref{eq:equil}) is perturbed by
an infinitesimal coherent structure $\delta Z$. For an S3T unstable structure, the induced flux divergence
tends to enhance the coherent structure $\delta Z$ producing the positive feedback required for instability.
So the goal of this section is to illuminate the eddy-mean flow dynamics leading to this positive feedback
and to understand qualitatively why the homogeneous equilibrium is more stable for small
and large values of $\tilde\beta$.




For zonal mean flows (\ref{eq:Q_evo2}), (\ref{eq:cov_evo2}) are simplified to:
\begin{equation}
\partial_t U=-\partial_y\left<u'v'\right>-rU=\partial_{y}\left(\partial_{y_2x_1}^2
\Delta_1^{-1}C\right)_{{\bf x}_1={\bf x}_2}-rU,\label{eq:U}
\end{equation}
and
\begin{equation}
\partial_t C=(A_1+A_2)C+\Xi,\label{eq:cov_evo3}
\end{equation}
where
\begin{equation}
A_i=-U_i\partial_{x_i}-(\beta-\partial_{y_iy_i}^2U)\Delta_i^{-1}\partial_{x_i}-r,
\end{equation}
respectively. As a result the zonal mean flow is driven by the momentum flux divergence of
the eddies. The perturbation in vorticity covariance
$\delta C$ that is induced by the mean flow perturbation $\delta U$ can be estimated immediately by assuming
that the system (\ref{eq:U})-(\ref{eq:cov_evo3}) is very close to the stability boundary, so that the growth rate is
small. In this case the mean flow evolves slow enough that it remains in equilibrium with the eddy covariance,
that is $d\delta C/dt\simeq 0$. \cite{Bakas-Ioannou-2012} showed that the ensemble mean momentum flux induced by
an infinitesimal sinusoidal mean flow
perturbation $\delta U=\epsilon\sin(my)$, where $\epsilon\ll 1$ (i.e the eigenfunction of (\ref{eq:Lc})), is
equal in this case to the integral
over time and over all zonal wavenumbers of the responses to all point excitations in the $y$ direction:
\begin{equation}
\delta\left<u'v'\right>=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
\int_0^\infty\overline{u'v'}(t)dt d\xi dk,\label{eq:uv_infty}
\end{equation}
where
$\overline{u'v'}(t)$ is the momentum flux at time $t$ produced by:
\begin{equation}
G(k,y-\xi)=B(k)h(y-\xi)e^{ikx+il_0(y-\xi)}.\label{eq:ini_vort}
\end{equation}
The Green's function $G$ has the form of a wavepacket with an amplitude $B(k)$ and a carrier wave with wavenumbers
$(k,l_0)$ that is modulated in the $y$ direction
by the wavepacket envelope $h(y)$. The characteristics of the amplitude, the wavenumber and the envelope depend on the
forcing characteristics, but in any case the calculation of the ensemble mean momentum fluxes is reduced
to calculating the momentum fluxes over the life cycle of wavepackets that are initially at different latitudes
and then adding their relative contributions.


As the wavepacket propagates in the latitudinal direction, its meridional wavenumber and frequency are going to
change due to shearing by the mean flow and due to the change of the mean vorticity gradient $\beta-U_{yy}$. The
resulting time variable momentum flux $\overline{u'v'}(t)$ can be calculated using ray tracing.
According to standard ray tracing arguments, the wave action is conserved along a ray (in the absence of dissipation)
leading to the momentum flux:
\begin{equation}
\overline{u'v'}(t)=-|B|^2A_M(t)e^{-2rt}|h(y-\eta(t))|^2,\label{eq:shear_flux}
\end{equation}
where $A_M(t)=kl_t/(k^2+l_t^2)^2$ is the momentum flux of the carrier wave that
determines the amplitude of the fluxes of the wavepacket and $l_t$, $\eta(t)$ are the time dependent meridional
wavenumber and position of the wavepacket respectively \citep{Andrews-87}.
Because of the small amplitude of the mean flow perturbation $\delta U$, the
wavenumber and position of the packet vary slowly on a time scale $O(\epsilon t)$
compared to the dissipation time scale $1/r$ and the dominant contribution to the time integral
in (\ref{eq:uv_infty}) comes from small times. We can therefore seek asymptotic solutions of the form
\begin{equation}
l_t=l_0+\epsilon l_1+\cdots~,~~~~\eta(t)=\xi+c_0t+\epsilon\eta_1(t)+\cdots ~,\label{eq:l_eta_as}
\end{equation}
where $c_0=2\beta kl_0/(k^2+l_0^2)^2$ is the group velocity in the absence of a mean flow and
calculate the integral of $\overline{u'v'}(t)$ over time from the leading order terms.
Substituting (\ref{eq:l_eta_as}) in (\ref{eq:shear_flux}) we obtain:
\begin{align}
\overline{u'v'}(t)&=\underbrace{-|B|^2A_M(0)e^{-2rt}|h(y-\xi-c_0t)|^2}_
{\overline{u'v'}_R} - \nonumber\\
 & \quad \underbrace{-
\epsilon |B|^2\left(\frac{dA_M}{dl_t}\right)_{l_0}l_1(t)e^{-2rt}|h(y-\xi-c_0t)|^2}_
{\overline{u'v'}_S}\nonumber\\
&\quad -\underbrace{\epsilon |B|^2A_M(0)\eta_1(t)e^{-2rt}\frac{d}{dy}|h(y-\xi-c_0t)|^2}_
{\overline{u'v'}_\beta}.\label{eq:shear_flux2}
\end{align}
The first term, $\overline{u'v'}_R$, arises from the momentum flux produced by a wavepacket in the absence
of a mean flow. Because $A_M(0)=kl_0/(k^2+l_0^2)^2$ is odd with respect to wavenumbers, this term does not
contribute to the ensemble averaged momentum flux when integrated over all wavenumbers and will be hereafter
ignored. The second term, $\overline{u'v'}_S$, arises from the
small change in the amplitude of the flux $A_M$ over a dissipation time scale. The third term,
$\overline{u'v'}_{\beta}$, arises from the small change in the position of the packet $\eta$
compared to a propagating packet in the absence of a mean flow. To summarize, the infinitesimal
mean flow refracts the wavepacket due to shearing by the mean flow and due to the change of the 
mean vorticity gradient and slightly changes the amplitude of the fluxes as well as
slightly speeds up or slows down the wavepacket. The sum of these two effects will produce the induced
momentum fluxes.


\subsection{The limit of small scale wavepackets with a short propagation range}


In order to clearly illustrate the behavior of the eddy fluxes, we consider the limit of
$\tilde\beta=\beta L_f/r\ll 1$, where $L_f$ is the scale of the wavepackets and in addition we assume
that the scale of the mean flow, $1/m$, is much larger than the scale of the wavepackets $mL_f\ll 1$.
In this limit, the wavepackets are dissipated before
propagating far from the initial position and the effect of the change in the mean vorticity gradient is
higher order. As a result, \cite{Bakas-Ioannou-2012} show that $l_1$ and $\eta_1$ decrease monotonically
with time with rates independent of $\delta U_{yy}$ and proportional to the