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% Paper Title
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\vspace*{20pt}
\begin{center}
\Large\bf
  A Numerical Experiment on the Zonal Jet Formation \\
  in 2-D Turbulent Flow on a Beta-Plane
\end{center}\vspace*{10pt}


%=========================================================================
% Author(s)
%=========================================================================


\begin{center}
\large\bf
  Shigeo Yoden \ and \ Jitsuko Hasegawa
\end{center}


%=========================================================================
% Affiliation(s)
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\begin{center}
  {\it Department of Geophysics, Kyoto University}
\end{center}\vspace*{10pt}


%=========================================================================
% Abstract
%=========================================================================


\begin{center}
\begin{minipage}{0.8\textwidth}
A series of numerical experiments on two-dimensional decaying
turbulence are performed for a non-divergent barotropic fluid on a
$\beta$-plane (a tangential plane to a rotating planet with the effect
of latitudinally varying Coriolis parameter $f$) in order to
survey the nature of zonal jet formation from random initial
fields. A parameter $\beta \equiv df/d\phi$ ($\phi$: latitude) is
swept systematically to show the dependence of latitudinal structure
of the zonal jet on its magnitude. Dynamics of such zonal jet
formations is investigated with a weakly nonlinear Rossby wave-zonal
flow interaction theory.
\end{minipage}
\end{center}\vspace*{10pt}


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% Body
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\section{Introduction}
%
An interesting feature of two-dimensional turbulence, in contrast to
ordinary three-dimensional turbulence, is the emergence of isolated
coherent vortices found by McWilliams~\cite{McW84,McW90}. Formation of
such coherent vortices is due to a self-organizing mechanism involved
in the disorder two-dimensional motion. As an application to
geophysical fluid, Rhines~\cite{R75} firstly studied the
two-dimensional turbulence on a $\beta$-plane, which is a tangential
plane to a rotating planet with the effect of latitudinally varying
Coriolis parameter~\cite{Pedl87}, to understand the effect of
differential rotation of the planet on the two-dimensional turbulence. 
His numerical result showed anisotropic growth of flow field and
emergence of alternating bands of mean zonal flow. These features due
to the $\beta$-effect have been reconfirmed in recent numerical
studies with much higher resolution models~\cite{MV91,VM93}.  They
noticed the zonal jet structures persist over many eddy turnover
times. Extremely persistent zonal jets are also observed in
quasigeostrophic two-layer $\beta$-plane turbulence forced by an
imposed unstable vertical shear~\cite{Panetta93}.

Williams~\cite{W78} took account of the exact spherical geometry in
his numerical experiment on forced two-dimensional turbulence, aiming
at a reproduction of the zonal band structure of Jovian atmosphere due
to the $\beta$-effect. He obtained a clear band structure of zonal
flow for a stochastic vorticity forcing. However, both longitudinal
periodicity and equatorial symmetry were assumed in his experiment to
reduce the computational domain to 1/16 of the entire sphere.
Advancement of computing facilities in 1990's enabled us to do these
computations with higher resolutions in full spherical
%geometry~\cite{YY93,CP96,NY97a,NY97b,HR98,YIHY99,IYHY00}.
geometry [9-15].
In the decaying turbulence experiments~\cite{YY93,YIHY99}, the
spontaneous formation of the zonal jets was found in mid- and
low-latitudes for strong rotation cases, in addition to the emergence
of a westward circumpolar vortex in high-latitudes. Such formation of
persistent zonal jets was also observed in the forced turbulence
experiment~\cite{NY97a,HR98}.

Pattern formation in two-dimensional turbulence on a rotating sphere
may be applied for understanding some phenomena in the real
geophysical fluids. The formation of zonal jets from a random flow
field has been considered as a mechanism to explain the band
structures in the atmospheres on the giant outer planets such as
Jupiter, Saturn, Uranus and Neptune~\cite{W78,CP96b}. Westward zonal
flow in high latitudes in the sun~\cite{DS90} has superficial
similarity to the circumpolar vortex obtained in our
experiments~\cite{YY93,NY97a}. Recently numerical study on the solar
tachocline was also done by investigating freely evolving stratified
turbulence in a thin rotating spherical shell~\cite{M01}.  Annular
variability observed in both hemispheres of the
atmosphere~\cite{TW98,TW00} is a modulation in the strength of the
circumpolar vortex in the troposphere and lower stratosphere with
intraseasonal and interannual time scales. Variations of the mean
zonal jets with significant barotropic components have much connection
with time variations of meridional and vertical potential vorticity
(PV) flux~\cite{H00}, and key dynamics of the annular variability may
be identical with that of the formation of the present circumpolar
vortex~\cite{R98}.

Rhines~\cite{R94} described fundamental mechanisms for the formation
of zonal jets on rotating spheres in some circumstances. One example
which is associated with the present subject is random external
stirring of fluid which becomes organized by the $\beta$-effect into
jets. The role of Rossby waves in angular momentum redistribution may
be important after full nonlinear upward cascade of turbulent energy
in small scales~\cite{HIYY00}.

In this study, we perform a series of numerical experiments on
two-dimensional decaying turbulence for a non-divergent barotropic
fluid on a $\beta$-plane in order to survey the nature of zonal jet
formation from random initial fields. We choose a $\beta$-plane
geometry with doubly periodic boundary conditions, because of simpler
dynamical situation than rotating sphere. Latitudinal dependence of
the $\beta$-effect and spherical geometry introduce some difficulty to
understand the dynamics, although these characteristics are more
relevant to the atmospheres.



\section{Numerical Model and Experiments}
%
Freely-evolving two-dimensional non-divergent flow on a $\beta$-plane
is governed by the following nondimensionalized vorticity equation:
\begin{equation}
       \frac{\partial \zeta}{\partial t}
   + u \frac{\partial \zeta}{\partial x} 
   + v \frac{\partial \zeta}{\partial y} 
   + \beta v 
   = - \nu_2 \Delta^2 \zeta,                          \label{eq:vorticity}
\end{equation}
where
$\zeta(x, y, t)$ is the vertical component of vorticity
($\equiv \partial v /\partial x - \partial u /\partial y$),
$(u, v)$ horizontal velocity ($\equiv (dx/dt, dy/dt)$),
$(x, y)$ eastward and northward coordinates, 
$t$ the time, $\Delta$ the horizontal Laplacian,
$\beta$ nondimensionalized beta ($f = f_0 + \beta y$),
and $\nu_2$ nondimensionalized hyperviscosity coefficient.
As $\nu_2 \rightarrow 0$, this equation gives the conservation 
low of PV (in the present case it is equivalent to absolute vorticity)
$q \equiv \zeta + f$ following Lagrangian motion.
If we introduce a streamfunction $\psi(x, y, t)$ 
($u=-\partial \psi /\partial y, v=\partial \psi /\partial x$), 
Eq.(\ref{eq:vorticity}) can be written as
\begin{equation}
   \frac{\partial}{\partial t} \Delta \psi
   + J(\psi, \Delta \psi) 
   + \beta \frac{\partial \psi}{\partial x}
   = - \nu_2 \Delta^3 \psi,                           \label{eq:vorticity-psi}
\end{equation}
where $J(A,B)$ is the horizontal Jacobian. As for the boundary
condition, we assume a periodic boundary condition in both directions
with $2\pi$: $\psi(x,0,t)=\psi(x,2\pi,t)$ and
$\psi(0,y,t)=\psi(2\pi,y,t)$.

A spectral model of Eq.(\ref{eq:vorticity-psi}) is constructed by a
double Fourier expansion ($\propto \exp[i(kx+ly)]$) of $\psi$ with the
truncation wavenumber $k_m = l_m = 1024$. A spectral transform
method~\cite{WP86} is used to compute the nonlinear Jacobian term;
grids for the transformation is 4096$\times$4096. Time evolution of
$\psi(x, y, t)$ is computed from an initial random flow field, of
which one-dimensional kinetic energy spectrum for a scalar wavenumber
$\kappa$ is given by
\begin{equation}
   E(\kappa,t\!=\!0)
   = \frac{A \kappa^{\gamma/2}}{(\kappa+\kappa_0)^{\gamma}}, 
\end{equation}
with $\gamma=1000$ and $\kappa_0 =226$. A constant $A$ is set so that
the total kinetic energy is 1/2 at $t=0$. Time integrations are done
until $t=12$ by the fourth-order Runge-Kutta method with the time
increment of $\Delta t = 4 \times 10^{-4}$. Ten values of $\beta$ are
taken as an experimental parameter: $\beta = 2^{n-1} \times 100 \
(n=1, 2, \cdots, 10)$. Even for the smallest value of $\beta = 100$, a
characteristic wavenumber $k_{\beta} = \sqrt{\beta / U}$ with $U \sim
1$, which gives a scale of transition from turbulence to Rossby-wave
motion, is much larger than the lowest wavenumber 1. Large value of
$k_{\beta}$ supports the soundness of the periodic boundary condition. 
The hyperviscosity coefficient is set to $\nu_2 = 1 \times
10^{-10}$. All of the computations are done in double precision.



\section{Results}
%
\begin{figure}[bt]
\begin{center}
\scalebox{.5}{\includegraphics{F1.ps}}
%\scalebox{.5}{\includegraphics*[viewport=10 0 750 540,clip]{F1.ps}}
\begin{minipage}{0.8\textwidth}
\caption[]{Time evolution of the streamfunction $\psi(x,y,t)$ and the
zonal mean zonal flow $[u](y,t)$ for $t=0$ (a), 2 (b), 8 (c), 
and 12 (d) in the run of $\beta=100$.
Negative contours of $\psi$ are drawn with broken lines.}
\end{minipage}
\end{center}
\label{fig1}
\end{figure}

Figure 1 shows time evolution of the streamfunction $\psi(x,y,t)$ and the
zonal mean zonal flow $[u](y,t)$ for the run of $\beta=100$.
Here $\displaystyle [a] = \frac{1}{2\pi} \int_0^{2\pi} a \, dx$.
Initially the flow field is isotropic, but vortices become elongated in
zonal direction. By $t=8$, the zonal wavenumber 1 component dominates, and
typical Rossby-wave motion can be seen in the animations of the streamfunction
field. At the same time, the zonal mean zonal flow develops, and its
latitudinal scale becomes large. The meridional profile of $[u]$ looks 
similar between $t=8$ and 12, which is indicative of the persistence of the
zonal mean zonal flow.


\begin{figure}[bt]
\begin{center}
\scalebox{.5}{\includegraphics{F2.ps}}
\begin{minipage}{0.8\textwidth}
\caption[]{Time evolution of two-dimensional energy spectrum for $t=0$
(a), 2 (b), 8 (c), and 12 (d). Note the wavenumber ranges and tone
values are different between (a,b) and (c,d). Broken lines give the
``dumbbell'' of anisotropic wave-turbulence boundary in wave vector
space. See text for details.}
\end{minipage}
\end{center}
\end{figure}


Time evolution of two-dimensional energy spectrum is shown in Fig.2
for the same run of $\beta=100$. Upward energy cascade to lower
wavenumbers takes place initially, and the spectrum is roughly
dependent only on the scalar wavenumber $\kappa=\sqrt{k^2 + l^2}$ at
$t=2$, indicating isotropic flow field in small scales. However, it is
not isotropic in large scales.  The two-dimensional spectrum shows
anisotropic distribution in low wavenumbers as pointed out by Vallis
and Maltrud~\cite{VM93}. Broken lines in Fig.2 give the ``dumbbell''
of anisotropic wave-turbulence boundary in wave vector space
introduced by them:
\begin{equation}
   (k,l) = k_{\beta} (\cos^{3/2} \theta, \cos^{1/2} \theta \sin \theta),
   \qquad \theta = \tan^{-1} (l/k).
\end{equation}
Note that the largest components of the spectrum are inside the
boundary, although the low-energy region shows a dumbbell shape in it. 
These are consistent with the findings in the spherical geometry
experiments~\cite{NY97b,HR98}.


Parameter dependence on $\beta$ is summarized as the time evolution of
one-dimensional energy spectrum(Fig.3). The spectral peak exists at a
lower wavenumber than $k_{\beta}$, and shifts to higher wavenumber as
$\beta$ increases. The one-dimensional spectrum shows a power low
roughly between the peak and $\kappa_0$, and the slope steepens as
$\beta$ increases: $E(\kappa) \propto \kappa^{\alpha}$, $\alpha \sim -
4$ for $\beta = 400$, while $\alpha \sim -5$ for $\beta = 12800$ at
$t=12$.  Energy density at the lowest wavenumbers is much smaller than
the peak value, indicating the influence of the periodic boundary
condition is weak in these experiments.

\begin{figure}[tbh]
\begin{center}
\scalebox{.65}{\includegraphics*[viewport=0 0 560 690,clip]{fig3.ps}}
\vspace{-1em}
\begin{minipage}{0.8\textwidth}
\caption[]{Time evolution of one-dimensional energy spectrum for 10
values of $\beta$. Vertical broken line gives $k_{\beta}$. 
Broken arrow on the abscissa gives the energy centroid wavenumber at
$t=12$, and solid one does that only for $k=0$ component.}
\end{minipage}
\end{center}
\label{fig3}
\end{figure}

In order to relate the present results with the real geophysical
fluid, we introduce a length scale of $L=5 \times 10^6$ m, which is
roughly the radius of the earth, and the velocity scale $U=10$
ms$^{-1}$. Then a dimensional value of beta becomes $\beta^* = (U/L^2)
\beta$, and $\beta^* = 4 \times 10^{-11}$ m$^{-1}$s$^{-1}$ for
$\beta=100$.  This value is comparable to that in mid-latitudes of the
earth. If we take $8L$ as a length scale and pick up only 1/8 in both
directions of $x$ and $y$, then the subset can be regarded as another
series of experiments with smaller value of beta $\beta^* = (U/L^2)
\beta /64$ for the same domain size with dimension.

Figure 4 shows meridional profiles the zonal mean zonal flow
$[u](y,t)$ for 16 values of $\beta^*$, with four values of overlapping
($\beta^*=4 \times 10^{-11} \sim 3.2 \times 10^{-10}$
m$^{-1}$s$^{-1}$).  Meridional scale of the mean zonal flow decreases
as $\beta^*$ decreases. The maximum value of $|[u]|$ is about 10
ms$^{-1}$, except for three cases of the largest $\beta$ in which
$k_{\beta}$ is not very different from $\kappa_0$.


\begin{figure}[h]
\begin{center}
\scalebox{.715}{\includegraphics*{fig4.ps}}
\begin{minipage}{0.8\textwidth}
\caption[]{Meridional profiles the zonal mean zonal flow
$[u](y,t)$ at $t=12$ for different value of $\beta$.
First 10 plots are enlargement of 1/8 part of the second 10 plots
including the maximum $|[u]|$.}
\end{minipage}
\end{center}
\label{fig4}
\end{figure}

Figure 5 summarizes the dependence of the meridional scale on $\beta^*$.
Energy centroid wavenumber for $k=0$ component is nearly parallel to
$k_{\beta}$ in a wide range of $\beta^*$; the meridional wavenumber
increases in proportion to $\sqrt{\beta^*}$. In the cases in which $k_{\beta}$
is not very different from $\kappa_0$, the proportionality breaks and the
meridional wavenumber becomes much smaller than $k_{\beta}$.


\begin{figure}[bt]
\begin{center}
%\scalebox{.5}{\includegraphics*[viewport=50 215 490 660,clip]{fig5.ps}}
\scalebox{.5}{\includegraphics{fig5.ps}}
\begin{minipage}{0.8\textwidth}
\caption[]{Energy centroid wavenumber (short broken line) and that
only for $k=0$ component (solid line) at $t=12$ for different value of
$\beta^*$ with dimension. Long broken line gives
$k_{\beta}^*$. Spectral peak at the initial state are also marked by
closed or open circles.}
\end{minipage}
\end{center}
\label{fig5}
\end{figure}



\section{Wave-mean flow interaction}
%
Development of the zonal mean jet is diagnosed with a weakly nonlinear
theory of interactions between Rossby waves and mean flow~\cite{HIYY00}.
We expand each dependent variable with a small parameter $\epsilon$:
\[
\begin{array}{rcrl}
   u     &=& U    +& \!\!\!\! \epsilon u' + \epsilon^2 u^{(2)} + \cdots , \\
   v     &=&       & \!\!\!\! \epsilon v' + \epsilon^2 v^{(2)} + \cdots , \\
   \psi  &=& \Psi +& \!\!\!\! \epsilon \psi'
                                      + \epsilon^2  \psi^{(2)} + \cdots , \\
   \zeta &=& Z    +& \!\!\!\! \epsilon \zeta'
                                      + \epsilon^2 \zeta^{(2)} + \cdots .
\end{array}
\]
Substituting these into Eq.(\ref{eq:vorticity}) and neglecting the viscosity
term, we obtain the following $O(\epsilon)$ vorticity equation:
\begin{equation}
       \frac{\partial \zeta'}{\partial t}
   + U \frac{\partial \zeta'}{\partial x}
   + \hat{\beta} v' = 0,                             \label{eq:vorticity-1}
\end{equation}
where $\hat{\beta} = \beta - \partial^2 U/\partial y^2$. $O(\epsilon)$
enstrophy equation is obtained by multiplying $\zeta'$ to
Eq.(\ref{eq:vorticity-1}):
\begin{equation}
       \frac{\partial}{\partial t}\frac{\zeta'^2}{2}
   + U \frac{\partial}{\partial x}\frac{\zeta'^2}{2}
   + \hat{\beta} v'\zeta' = 0,                       \label{eq:enstrophy-1}
\end{equation}
After some manipulation a wave activity equation is obtained as follows:
\begin{equation}
   \frac{\partial A}{\partial t} + \nabla \cdot \mbox{\boldmath $F$} = 0,
                                                     \label{eq:wave-activity}
\end{equation}
where the wave activity is given by $A=\zeta'^2/(2\hat{\beta})$ and
its flux by $\mbox{\boldmath $F$} = (UA + (v'^2 -u'^2)/2, -u'v')$.
On the other hand, 
zonal mean of the zonal momentum equation for $O(\epsilon^2)$ becomes
\begin{equation}
   \frac{\partial}{\partial t}[u^{(2)}] - [v'\zeta'] = 0.
                                                     \label{eq:zm_x_mom-2}
\end{equation}
Eliminating $[v'\zeta']=[\nabla \cdot \mbox{\boldmath $F$}]$ in 
Eqs.(\ref{eq:wave-activity}) and (\ref{eq:zm_x_mom-2}), we obtain 
the conservation law of pseudo-momentum:
\begin{equation}
   \frac{\partial}{\partial t} \left\{ [u^{(2)}] + [A] \right\} = 0.
                                                     \label{eq:pseudo_mom-2}
\end{equation}


\begin{figure}[bt]
\begin{center}
\scalebox{.5}{\includegraphics{F6.ps}}
\begin{minipage}{0.8\textwidth}
\caption[]{Diagnosis of wave-mean flow interactions for a run with a
low-resolution model ($\beta=200$). (a) time mean of the zonal mean
zonal flow $\overline{[u]}(y)$ for $10 \le t \le 20$, (b) time
evolution of the zonal mean zonal flow $[u](y,t)$, (c) time
evolution of $\partial [u]/\partial t$, (d) time mean of $\hat{\beta}
= \beta - d^2 \overline{[u]} /dy^2$, (e) time evolution of the
time-change of zonal mean wave activity $\partial [A]/\partial t$, and 
(f) time evolution of the divergence of wave activity flux
$[\nabla \cdot \mbox{\boldmath $F$}] = [v' \zeta']$.}
\end{minipage}
\end{center}
\label{fig6}
\end{figure}


Firstly, interactions between Rossby waves and the zonal mean zonal
flow are diagnosed after the development of the mean zonal flow in a
low-resolution model of $k_m=l_m=128$ with $\beta=200$. Figure 6(a)
shows the time mean of the zonal mean zonal flow $\overline{[u]}(y)$
for $10 \le t \le 20$, and (b) shows its time evolution
$[u](y,t)$. The zonal mean jets are well developed by $t=10$ and
persists robustly over the following period. Deceleration and
acceleration of the zonal mean jets propagate rather constantly in
either latitudinal direction as shown in (c). The time-change of the
zonal mean wave activity $\partial [A]/\partial t$ (e) almost
compensates $\partial [u]/\partial t$, and is associated with the
divergence of wave activity flux $\nabla \cdot \mbox{\boldmath $F$}$
(f). These plots indicate the usefulness of the wave-mean flow
interaction theory in diagnosing the time variation of the zonal mean
jets after their development.



\section{Concluding remarks}
%
The formation of zonal jets in two-dimensional decaying turbulence on
a $\beta$-plane is investigated in a wide parameter range of $\beta$.
Their latitudinal scale becomes small as $\propto 1/\sqrt{\beta}$.
Extremely persistent nature of the jets is reconfirmed in the present
study with a high-resolution model. A weakly nonlinear Rossby
wave-zonal flow interaction theory is a useful tool to diagnose the
time variation of the zonal mean jets after their development. It is
our next step to apply this tool to predict the latitudinal position
and timing of the zonal jet formation.


%=========================================================================
% Sample References
%=========================================================================


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\end{document}

