% Subject:      The IUTAM/IUGG Symposium, 
%               Developments in Geophysical Turbulence, June 16-19, 1998
%               Emergence of circumpolar vortex in two dimensional turbulence 
%               on a rotating sphere
%               Kluwer Academic Publishers
%
% History:      Y.-Y. HAYASHI 1998-09-28


\documentstyle[editedvolume,psfig]{Crckapb} 
\newcommand{\stt}{\small\tt}

\begin{opening}
\title{EMERGENCE OF CIRCUMPOLAR VORTEX IN TWO DIMENSIONAL TURBULENCE 
       ON A ROTATING SPHERE}

\author{Y.-Y. HAYASHI}
\institute{Division of Earth and Planetary Sciences, Hokkaido University\\
                Sapporo 060-0810, JAPAN}
\author{K. ISHIOKA}
\author{M. YAMADA}
\institute{Graduate School of Mathematical Sciences, University of Tokyo\\
                Tokyo 153-8914, JAPAN}
\author{S. YODEN}
\institute{Department of Geophysics, Kyoto University\\
                Kyoto, 606-8502, JAPAN}

\end{opening}

\runningtitle{2-D TURBULENCE ON A ROTATING SPHERE}

\begin{document}

\begin{abstract}
        
    Characteristics of the decaying non-divergent 
    two dimensional turbulence 
    on a rotating sphere is considered. 
    The spontaneous appearance of circumpolar easterly vortices 
    reported in the previous study 
    (Yoden and Yamada, 1993)
    is interpreted by the Rossby wave property from the framework of
    weak-nonlinear theory. 
    The resolution of the numerical 
    model utilized by Yoden and Yamada (1993) is T85  
    which is the resolution not enough to represent the upward energy 
    cascade of two dimensional turbulence. 
    The initial energy spectral peak adapted there is located 
    almost in the range of wave regime 
    especially when the rotation rate is large, 
    which justifies the explanation of angular momentum redistribution
    by Rossby waves. 
    A series of new experiments with the higher resolution T341 
    and new sets of initial energy spectral distributions  
    are performed to confirm that the circumpolar vortices 
    appear even after the full nonlinear 
    upward cascade of turbulent energy and 
    that the band structure of angular momentum emerges even when the 
    rotation rate of the system is large. 
\end{abstract}

\section{Introduction}

   It has been recognized that a two dimensional turbulent flow on a
	rotating sphere tends to have a zonally band structure, i.e., 
	bands of mean zonal flow with alternating flow direction. 
   The pioneering work of this issue is Williams (1978), where a time
	evolution of a randomly forced two dimensional nondivergent fluid 
	was numerically investigated to show that a zonally band structure
	emerges when the rotation rate and the radius of the sphere were
	adapted from those of Jupiter's. 
   However, the computational domain utilized by Williams (1978) was
	only 1/16 of the entire sphere. 
   The longitudinal periodicity and equatorial symmetry 
	were assumed. 

   A full spherical computation of two dimensional turbulence 
	on a sphere was rather recently carried out by 
   	Yoden and Yamada (1993, hereafter referred to as YY93). 
   They investigated decaying, in stead of forced, 
	two dimensional nondivergent turbulent flow on a rotating
	sphere with the resolution of T85 
	(the triangular spectral truncation of spherical harmonics 
	at the total wavenumber 85, that is, 
	256 (longitude)$\times$ 128 (latitude) grids).  
   YY93 found that
        an easterly circumpolar vortex tends to emerge 
        spontaneously 
        in a decaying turbulent flow on a rotating sphere
        especially when the rotation rate is high. 
   
   A full spherical version of the forced turbulence was 
	investigated quite recently 
	by Nozawa and Yoden (1997a, b) 
	in the same way as Williams (1978) 
	but with the resolution T199 (600 $\times$ 300 grids). 
   They reconfirmed the spontaneous formation of the band structure
	of the zonal flow.
   Moreover,       
	they found that, similarly to the decaying turbulence cases, 
        an easterly circumpolar vortex tends to appear. 
   The property of the forced nondivergent turbulent flow 
	on a rotating sphere may be understood through the property 
	of the decaying turbulent flow. 

   However, there remains a concern about the results of YY93. 
   As will be described below,  
	the flows obtained by YY93 are not very turbulent
	especially when the rotation rate of the system is large. 
   The initial condition used in YY93 
	contains, when the rotation rate is large, 
	a large part of energy 
	in the large scales where the linear rotation term dominates 
	the nonlinear inertial term. 
   On the other hand, 
	YY93 shows that the circumpolar easterly flow appears clearer 
	as the increase of the rotation rate. 
   The results might be altered 
	when the resolution of the model is increased to 
	to represent the full nonlinear upward energy cascade 
	of two dimensional turbulence. 

   In the followings, 
	we will describe 
	the formation of circumpolar easterly vortex 
	in YY93 
	from the framework of weak nonlinear Rossby wave property 
	on a sphere. 
   We will then demonstrate some of 
	the results of further numerical experiments 
	with a high-resolution model (T341 = 1024 $\times$ 512)
	to see the pattern formation due to the real 
	inverse cascade from small scales
	given by some appropriate initial conditions. 
   We will confirm the tendency of the formation of 
	circumpolar easterly vortex even when the rotation rate is high,
	in addition to the spontaneous formation of the band structure 
	of zonal flow. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nondivergent Two Dimensional Flow on a Sphere}

   Freely-evolving two dimensional nondivergent flow 
	on a rotating sphere is governed by the following 
	vorticity equation:
	\begin{equation}
	\label{eq:vor0}
	   \frac{\partial \zeta}{\partial t} 
	 + J(\psi, \, \zeta)
	 + 2 \Omega
	\frac{\partial \psi}{\partial \lambda}
	 = (-1)^{p+1} \nu_{2p} \left(\Delta +2\right)^{p} \zeta,
	\end{equation}
	where $\psi(\lambda, \mu, t)$ is a streamfunction field,
	$\zeta(\lambda, \mu, t)$ vertical component of vorticity 
	($\equiv \Delta \psi$), 
	$\lambda$ longitude, 
	$\mu=\sin\phi$ sine latitude, $t$ time,
	$J(A,B)$ horizontal Jacobian, 
	$\Delta$ horizontal Laplacian.
   On the right hand side, 
	hyper-viscosity of the order of $p$ is 
	placed with a viscosity coefficient $\nu_{2p}$
	for the use of numerical calculation presented later. 
   Note that the physical quantities are nondimensionalized  
	by the radius of the sphere $a$ as the length scale,
	   the rotation rate of the system $\Omega^*$ as the time scale, 
	and the energy velocity scale $U$ as the advection velocity scale. 
   By this scaling, 
	we have to consider only 
	the motion with unit initial total energy 
		on the rotating sphere with the unit radius 
		and the nondimensionalized rotation rate 
		$\Omega \equiv U \Omega^*/a $, or, 
		the reciprocal of Rossby number. 

   The important quantity 
	which characterizes two dimensional nondivergent flow 
	on a rotating sphere is 
	the transition wavenumber
	that is defined as a local wavenumber at 
	which the magnitude of the nonlinear term is 
	comparable to that of the linear planetary vorticity 
	advection term (so called $\beta$ term)
	(Rhines 1975, Vallis and Maltrud, 1993, Rhines 1994). 
   For the system given by (\ref{eq:vor0}), 
	the transition wavenumber may be written as 
	$%\begin{eqnarray}
                k_{\beta} = \sqrt{2 \Omega \cos \phi},
        $%\end{eqnarray}
	where $\phi$ is the latitude. 
   The interpretation of the transition wavenumber is that
	turbulence dominates when the characteristic wavenumber 
	of the flow is larger than the transition wavenumber, 
	while Rossby waves dominate when it is smaller. 
   On a rotating sphere, 
	the transition wavenumber vanishes at the pole, 
	while it is maximum at the equator, 
	since the $\beta$ effect, i.e., 
	the linear planetary vorticity advection, 
	is most effective at the equator. 
   This implies that, 	
	for a disturbance with a given characteristic scale,
	flow becomes turbulent more easily 
	in the polar region than in the lower latitudes. 
   Provided that the characteristic wavenumber of the flow is
 	$\tilde{n}$, 
	there appears a Rossby wave dominant region
	around the equator between the latitudes 
	$\phi_t = \pm \cos^{-1}(\tilde{n}^2/2\Omega)$
	when the rotation late is sufficiently large. 

   In the numerical experiments of YY93,  
	the value of $\Omega$ ranges from 0 to 400. 
   The corresponding transition wavenumber ranges from 0 to 28 
	at the equator. 
   Since the initial energy spectrum of YY93 is given by
	$
	   E(n,t\!=\!0)=A n^5/e^{-n/2}, 
	$
	where $n$ is the total wavenumber, 
	the energy spectral peak is located at $n=10$ 
	(Fig.\ref{fig:E-init}), 
	and the enstrophy spectral peak is located at $n=12$.
   Thus for $\Omega > 50 \sim 100$, there appears a latitudinal 
	regions where Rossby waves dominate in the lower latitudes. 
   For the rapid rotation case $\Omega = 400$, 
	the transition latitude from wavy to turbulent regime becomes 
	$\phi_t = \pm 83^{\circ}$,
	provided that $\tilde{n}=10$. 
   In this sense, the numerical flows presented by YY93 are 
        actually not 
	very turbulent especially when the rotation rate 
        is large. 


{\small
\begin{figure}
\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/fig1r.ps,height=8.0cm} 
\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	Five initial energy spectra. 
	The spectral form used by YY93 is denoted by yy1993.
	All the other spectral forms are given by eq.(2).  
	Five vertical dash-lines indicate the global mean 
	transition wavenumber 
	$n_{\beta} \equiv \sqrt{<k_{\beta}> /2\sqrt{2}} = 
	\sqrt{\pi\Omega /(4\sqrt{2})}$ 
	defined by Nozawa and Yoden (1997a), 
	where $<\hspace{1ex}>$ denotes global mean 
	and the numerical factor $2\sqrt{2}$ is retained
	following their definition.
	}
\label{fig:E-init}
\end{figure}
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weak Nonlinear Theory}

   The formation of the easterly circumpolar vortex 
	found by YY93 may be considered as the result of 
	turbulent mixing of the potential vorticity 
	in the polar region 
	where the flow regime is regarded as turbulent. 
   On a rotating sphere, 
	mixing of the higher potential vorticity around the pole
	and the lower potential vorticity of the lower latitudes 
	causes easterly flow in the polar region,  
	provided that there is no net positive angular momentum 
	supply from the outside of the polar mixed region. 
   When the rotation rate is high and 
	the outside of the polar mixed  
	region can be regarded as in the Rossby wave regime,
	it can be demonstrated that the net angular momentum 
	supply from the outside of the polar mixed region
	is negative (easterly). 
   As is expected from the knowledge that 
	the pseudo angular momentum associated with Rossby wave 
	is negative, 
	the issue is to find out whether the polar mixed region 
	absorbs Rossby waves from the wavy region 
	to gain easterly momentum or
	radiates Rossby waves toward the wavy region 
	to gain westerly momentum. 

   Let us consider the weak nonlinear version of 
	the basic system (\ref{eq:vor0}) to investigate
	the wave mean-flow interaction of 
	Rossby waves on a sphere. 
   The amplitude expansion around a zonal mean basic state 
	$\overline{u}(\phi)$
	gives the first order linearized equation 
	as
	\begin{eqnarray}
	\label{eq:vor1}
	&&
	   \frac{\partial \zeta'}{\partial t} 
	 + \frac{\overline{u}}{\cos\phi}
	   \frac{\partial \zeta'}{\partial \lambda} 
	 + \frac{\hat{\beta}}{\cos\phi}
	   \frac{\partial \psi'}{\partial \lambda}
	 = f', 
	\\
	&&\hspace*{1cm} 
	\hat{\beta} \equiv 2\Omega\cos\phi 
			- \frac{\partial }{\partial \phi} 
			  \left[
				\frac{1}{\cos\phi}
			    	\frac{\partial }{\partial \phi} 
				\cos\phi \overline{u}
			  \right] ,
	\nonumber
	\end{eqnarray}
	where $f'$ is the viscous dissipation term,  $'$ and 
	$\bar{\hspace*{1ex}}$  
	denote the first order and the zonal mean, respectively.   
   The second order conservation law of the first order quantity
	can be derived immediately from (\ref{eq:vor1}):
	\begin{eqnarray}
	\label{eq:vor2}
	&&
	   \frac{\partial }{\partial t} 
		{\cal A}
	 + \nabla \cdot {\cal F}	
	 = \frac{\cos\phi}{\hat{\beta}}\zeta'f'
	\\ && \hspace*{1cm}
	{\cal A} \equiv
		\frac{\zeta'{}^2}
		     {2}
		\frac{\cos\phi}{\hat{\beta}}
	\nonumber
	\\ && \hspace*{1cm}
	{\cal F} \equiv
	   + \left(\overline{u} {\cal A} 
		+ \frac{1}{2}
		  \cos\phi (v'{}^2 - u'{}^2), 
		  \cos\phi u'v'	
	     \right), 	
	\nonumber
	\end{eqnarray}
	where $u'$ and $v'$ is the first order velocity component,
	and $\nabla\cdot$ is spherical divergence operator. 
   The angular momentum conservation law is more useful 
	as the second order zonal mean equation:
	\begin{eqnarray}
	\label{eq:am2}
	   \frac{\partial }{\partial t} 
		(\overline{u^{(2)}}\cos\phi)
	 - \overline{v'\zeta'} \cos\phi
	 = \overline{f^{(2)}},
	\end{eqnarray}
	where $f^{(2)}$ is the viscous dissipation term.
   Combined with the zonal mean of (\ref{eq:vor2})
	with the relation 
		$
		\overline{\nabla \cdot {\cal F}}
		= \cos \phi \overline{v'\zeta'}
		$, 
   (\ref{eq:am2}) yields   	
	\begin{eqnarray}
	\label{eq:am3}
	   \frac{\partial }{\partial t} 
	   \left[
		\overline{u^{(2)}}\cos\phi
	 	+ \overline{\cal A}
	   \right]
	 = \cos\phi \overline{f^{(2)}}
	  + \frac{\cos\phi}{\hat{\beta}}\overline{\zeta'f'}.
	\end{eqnarray}

   The important character of the Rossby waves propagating on 
	a sphere is that the second order conserved quantity
	associated with a wave packet 
	is $\cal A$ as shown in (\ref{eq:vor2}), 
	and the related mean quantity is easterly 
	angular momentum as shown in(\ref{eq:am3}). 
   When the rotation late is range and 
	the effect of $\overline{u}$ can be neglected,     
   WKBJ theory of Rossby wave propagation on a sphere 
	(Hoskins and Karoly, 1981; Hayashi and Matsuno 1984) 
	shows that 
	a Rossby wave packet propagates along a great circle 
	seeing from the frame of rotation with the speed of
	eastward phase velocity of the wave packet. 
   Let us assume a wave packet whose pitch angle 
	at the equator 
	is sufficiently large so that 
	it can propagate into the higher latitudes where $\beta$ is
	small. 
   The propagation of Rossby wave packet induces 
	the second order easterly flow $\overline{u^{(2)}}$, 
	whose magnitude is enhanced in the higher latitude
	because of the factor $\cos\phi$, 
	since the angular momentum associated with the wave packet
	is conserved during its propagation. 
   If we allow the wave mean-flow interaction, 
	we can expect that 
	the large easterly fluctuation of zonal mean flow 
	in the higher latitudes 
	enhances the critical latitude absorption there, 
	which contributes the easterly angular momentum accumulation
	in the higher latitudes. 

{\small
\begin{figure}
%\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/wn.ps,height=10.0cm} 
%\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	The comparison between the nonlinear (upper left)
	and weak nonlinear (upper right) evolutions 
	of the angular momentum
	starting from the same initial value 
	for the case of $\Omega=400$.
	Also shown are $\overline{\cal A}$ (lower left) and 
	$\hat{\beta}$ (lower right) of the nonlinear calculation.
	The numerical model utilized is the same as YY93 (T85)
	and hence, these figures except for the upper right represent 
	one of the realizations in YY93 $\Omega=400$ cases. 
	}
\label{fig:weak}
\end{figure}
}

   In order to confirm the weak nonlinear scenario mentioned above, 
	we have carried out a numerical comparison
	between the nonlinear and weak nonlinear evolutions 
	starting from one of the same initial values
	with the same T85 model utilized by YY93.  
   The weak nonlinear system is composed of eqs. (\ref{eq:vor1}) 
	and (\ref{eq:am2}) but with 
	$\overline{u^{(2)}}$ replaced by $\overline{u}$ in 
	(\ref{eq:am2}); $\overline{u}$ is now a function of $t$ and
	$\phi$. 
   Fig.\ref{fig:weak} shows a typical example of the comparative 
	result where the accumulation of the easterly angular momentum 
	in the polar region occurs very similarly. 
   Once the easterly acceleration starts in the polar region, 
	it continues steadily.	
   The growth of easterly seems to be a little faster 
	for the weak nonlinear evolution. 

   In the experiments of YY93, the spatial distribution of the 
	amplitude of initial disturbance is statistically 
	uniform on the sphere. 
   When the rotation rate is large, 
 	since $\hat{\beta}\sim\beta=2\Omega\cos\phi$, 
	we have $\overline{\cal A} \sim \overline{\zeta'{}^2}/4\Omega$
	(Fig.\ref{fig:weak} lower panels). 
   This means that the initial distribution of $\overline{\cal A}$ is also 
	statistically uniform on the sphere,
   	which can be observed in Fig.\ref{fig:weak}(lower left). 
   Singularity appears only in the higher latitudes 
	corresponding to the regime transition 
	from the Rossby wave of the lower latitudes to the turbulent, 
	or, the occurrence of critical latitude absorption 
	and strong wave-mean flow interaction 
	from the weak nonlinear sense. 
   As time goes on, the amount of $\overline{\cal A}$ gradually decreases 
	except for the singular polar region
	and correspondingly 
	the westerly angular momentum increases slightly and uniformly 
	in the mid and lower latitudes. 

   According to our comparative calculations, 
	the weak nonlinear versions resemble
	the full nonlinear results of YY93 
	for the cases of $\Omega=400$ quite well, and
	$\Omega=200$ fairly well. 
   However, the cases with $\Omega=100$,
	the evolution of mean flow is occasionally different 
	in the mid and low latitudes.
   For the cases of smaller values of $\Omega=100$,
	there are almost no resemblance. 
   Remember that the appearance of circumpolar vortex is not very clear
	in the nonlinear calculation either. 


\section{Results with the Higher Resolution T341}

   As shown in the previous section, the experiments of YY93 are
	not very turbulent; 
	the appearance of the circumpolar vortex in YY93 cannot be
	regarded as a result of 
	full nonlinear upward energy cascade
	of two dimensional turbulence. 
   And also, as is evident from Fig.\ref{fig:weak} 
	there are not clear spontaneous evolution of 
	the band structure of zonal mean flow
   When the rotation rate is large ($\Omega=400$), 
	a strong development of zonal mean flow occurs only in the
	polar region. 
   In the mid and low latitudes, 
	the initial zonal mean flow does not change greatly,
	although latitudinally uniform westerly mean flow 
	acceleration occurs.  
	
   Now, we will present some of the results 
	obtained by the higher resolution model
	(T341 = 1024 $\times$ 512). 
   The experiments are performed 
	to confirm the emergence of the circumpolar vortex 
	and to observe the formation of band structure 
        even from the initial spectral profile 
        which does not contain spectral energy 
        in the Rossby wave regime. 

   The new initial energy spectrum is given by
	$
	   E(n,t\!=\!0)=A n^{\gamma/2}/(n+n_{0})^{\gamma}.
	$
   Following the normalization utilized by YY93, 
   $A$ is given by the condition
	$
	   \sum_{n=2}^{N} E(n,t\!=\!0)=1
	$
   (the total kinetic energy is unity).	
   The initial complex amplitudes of the spherical harmonics expansion
	of the stream function 
	are determined so that they have random amplitude and 
	phase under the above restriction.
   In Fig.\ref{fig:E-init}, four sets of the energy spectra 
	utilized in our experiments are presented. 
   The values of ($n_0$, $\gamma$) are
	(10,1000) for cpn010,
	(50,1000) for cpn050 (the standard set),
	(100,1000) for cpn100,
	and
	(50,100) for cpg100. 

   Ten initial conditions are made with different random
	numbers for each experimental set.  
   The hyper-viscosity is set with $p=8$ and 
	$\nu_{2p} = 1 \times 10^{-38}$.  
   Time evolutions are computed until $t=5$ 
	by the fourth-order Runge-Kutta method with the
	time increment of $\Delta t = 1 \times 10^{-3}$. 



{\small
\begin{figure}
\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/spview.ps,height=9.0cm} 
\vspace{-2cm}  % amount of vertical space needed
\caption{
	Vorticity fields at $t=5$ of the cpn050 experiment
	for $\Omega=0$ (left), $\Omega=50$ (center), and 
	$\Omega=400$ (right) with the T341 model. 
	Orthographic projection from 
	$\lambda=0^{\circ}$, $\phi=0^{\circ}$ is used and lines of
	meridians and parallels are drawn for every 30$^{\circ}$.  
	Solid line on the right side of each panel shows
	the zonal mean zonal angular momentum as a function of sine
	latitude.
	}
\label{fig:spview}
\end{figure}
}

   Fig.\ref{fig:spview} is an example of the vorticity fields 
	at $t=5$ evolved from the same initial condition of the 
	spectral type cpn050 but with the different value of 
	$\Omega$. 
   In the non-rotational case (left), 
	a number of coherent vortices appear and then 
	their population decreases gradually through mergers.
   The amount of zonal mean angular momentum plotted 
	in the right hand side of the panel
	may seem to be very large, 
	but the profile changes as the coherent vortices move.
   In the high-rotation case $\Omega=400$(right), 
	the vorticity field shows zonally elongated structures.  
   The profile of the zonal mean angular momentum 
	shows the emergence of easterly circumpolar vortex, 
	particularly in the southern hemisphere for this run.
   In addition to that, the zonal band structure appear 
	in the mid and low latitudes. 
   As will described below, the zonal band structure of $\Omega=400$
	obtained by this T341 experiment is not just a remainder 
	of the initial zonal profiles as that of T85 experiment 
	of YY93.

{\small
\begin{figure}
\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/zmam0r.ps,height=8.0cm} 
\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	Dashed lines represent 
	the zonal mean angular momentum distributions of the 10 
	initial conditions of cpn050 (left) and 
	their final profiles at $t=5$ (right) for $\Omega=400$ 
	with the T341 model. 
	Thick lines represent the ensemble average of 
	those dashed lines.
	}
\label{fig:zm0-5}
\end{figure}
}

   Fig.\ref{fig:zm0-5} shows the statistical feature of 
	the latitudinal distributions of zonal mean angular momentum.
   10 initial conditions of cpn050 and 
	those at $t=5$ (right) for $\Omega=400$ are plotted. 
   It is clearly observed that 
	the circum polar easterly vortex appears
	statistically
	through the nonlinear energy upward cascade 
	from the initial spectral peak located at wavenumber 50. 
   The latitudinal distribution of 
	the zonal mean angular momentum of each realization 
	is different to each other.
   However, there seems to be a typical latitudinal
	wavenumber of zonal mean flow 
	roughly proportional to $n_\beta$
	(Fig.\ref{fig:zmam}).
   The characteristics observed in YY93 and 
	considered by the weak nonlinear theory 
	still appear even with this turbulent initial condition;
   Strong easterly appears in the high latitudes 
	while weak westerly appears in the mid and low latitudes.
   This tendency can be more clearly observed 
	in the ensemble average profile,
	where the rapid latitudinal variation of the 
	angular momentum profile of each realization
	is smoothed out.

{\small
\begin{figure}
%\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/zmall.ps,height=9cm} 
%\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	Dependence of zonal mean angular momentum on
	   the rotation rate $\Omega$ for the five sets
	   of the different initial spectral profiles. 
	   Ensemble average of 2$\times$10 runs 
	   under the assumption of
	   equatorial symmetry.
	}
\label{fig:zm-all}
\end{figure}
}

   Fig.\ref{fig:zm-all} shows the dependence of 
	the zonal mean zonal angular momentum at $t=5$ on $\Omega$ 
	for the five experimental sets with 
	the different initial spectral profiles. 
   The formation of the easterly circumpolar vortex for large 
	$\Omega$ is a robust result independent of the choice 
	of the initial energy spectrum.  
   As $\Omega$ increases, 
	the intensity of the easterly vortex
	becomes large and its maximum position shifts toward 
	the higher latitudes.  
   Note that the easterly vortex for $\Omega=400$ is 
	stronger for the larger value of $n_{0}$ 
	from the set cpn010 to cpn100. 
   As a result of the ensemble average, 
	westerly flow appears in the lower latitudes 
	for large $\Omega$ cases. 
   However, as mentioned previously 
	each realization shows a zonal band structure in mid and
	low latitudes.  
   The ensemble average masks the band structure 
	because the latitudinal position of the jets 
	dependent rather randomly on each realization.


{\small
\begin{figure}
%\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/zmam.eps,height=9.0cm} 
%\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	The time evolutions of 
	   zonal mean angular momentum 
	   from the same initial condition of cpn050
	   but for the different values of $\Omega$. 
	}
\label{fig:zmam}
\end{figure}
}

   Fig.\ref{fig:zmam} shows the time evolution 
	of the zonal mean angular momentum for various 
	values of $\Omega$
	starting from the same initial value 
	of the energy spectral form cpn050. 
   The band structure of the zonal angular momentum 
	can be clearly observed as the increase of
	$\Omega$. 
   The development of the band structure occurs rather
	early stage of the time evolution. 
   At the beginning of the integration, 
	the zonal angular momentum profile has
	a rather high 
	characteristic latitudinal wavenumber,
	since the initial spectral peak is located 
	at $n=50$. 
   The characteristic wavenumber evolves rapidly toward 
	the lower wavenumber.  
   Around $t\sim 0.5$, the band structure comes to a 
	steady or very slowly evolving state. 
   As reported by Nozawa and Yoden (1997a) for the forced
	two dimensional turbulence on a sphere,
	the characteristic wavenumber of the 
	zonal band profile seems to be roughly proportional 
	to $n_\beta$.

{\small
\begin{figure}
%\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/tczmam1r.ps,height=9.0cm} 
\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	The time evolution of 
	   zonal mean angular momentum for $\Omega=400$
	starting from 
	cpn010 (left), cpn050 (center), and cpn100 (right). 
	Note that the time scale is plotted by the log scale.
	}
\label{fig:tczmam1}
\end{figure}
}

   Fig.\ref{fig:tczmam1} shows the time evolution 
	of the zonal mean angular momentum for the case of 
	$\Omega=400$ 
	starting from the three different initial energy spectral 
	profiles cpn010, cpn050, and cpn100.
   Note that the time scale is different from
	Fig.\ref{fig:zmam}; 
	log time scale is adopted. 
   As is discussed in the previous section, 
	when the initial spectral peak is located in the spectral 
	range of Rossby wave regime ($n_0<k_\beta(\phi=0)$), 
	the zonal band structure is almost completely 
	determined by the initial zonal band profile 
	(cpn010, Fig.\ref{fig:tczmam1} left). 
   However, 
	when the initial energy profile has the spectral
	peak located in the turbulent regime 
	($n_0 \gg k_\beta(\phi=0)$), 
	a spontaneous evolution of the zonal band structure 
	occurs 
	(cpn050 and cpn100, 
	Fig.\ref{fig:tczmam1} center and right).
   Fig.\ref{fig:tczmam1} indicates clearly
	that the band structure establishes 
	as the spontaneous evolution of 
	the nonlinear upward cascade of energy. 

   The rapid change of the energy spectrum also occurs 
	at rather early stage of the time evolution
	(not shown). 
   This should be directly related 
	to the establishment of the band structure
	at around $t\sim 0.5$.
   The distribution of the energy spectrum is characterized by 
	the appearance of anisotropy.     
   The two dimensional energy spectrum at $t=5$ 
	is shown in Fig.\ref{fig:2dspctrum}.
   The energy density of small $m$ is dominant in the range of 
	$n<k_{\beta}(\phi=0)$ while the energy is very small 
	in an airfoil-shaped region at the lower edge in the
	wavenumber space $(m,n)$ 
	as found in the forced turbulence experiments 
	Nozawa and Yoden (1997a,b). 
   The existence of such a small energy region is 
        qualitatively explained by the analogy of a dumbbell region obtained
	in the $\beta$-plane experiment by Vallis and Multrad (1993). 
   Anisotropic distribution of the energy is also found in the high
	total-wavenumber region $n>k_{\beta}$; energy is confined in the
	region of small zonal wavenumber $m$.  

{\small
\begin{figure}
\vspace{-0.5cm}  % amount of vertical space needed
\centering \leavevmode 
\psfig{file=figs/2dspctrum.ps,height=9.0cm} 
\vspace{-1.5cm}  % amount of vertical space needed
\caption{
	Two dimensional energy spectrum at $t=5$ 
	of the cpn050 experiment 
	for $\Omega=0$ (left), $\Omega=50$ (center), and 
	$\Omega=400$ (right) with the T341 model. 
        The air foil region determined by Nozawa and Yoden (1997b)
	is denoted by thick line.
	}
\label{fig:2dspctrum}
\end{figure}
}



\section{Concluding remarks}

    The spontaneous appearance of circumpolar easterly vortices 
	reported by YY93 was interpreted by the Rossby wave property 
	from the framework of weak-nonlinear theory. 
    The initial energy spectral peak adapted there is located 
	almost in the range of wave regime 
	especially when the rotation rate is large, 
	which justifies the explanation of angular momentum 
	redistribution by Rossby waves. 
    A series of new experiments with the higher resolution T341 
    	and new sets of initial energy spectral distributions  
	are performed to give the following results:
\begin{enumerate}
\item The circumpolar easterly 
	vortices appear 
	in high latitudes and 
	the zonal band structures of alternating mean zonal
	jets appear in mid and low latitudes 
	even after the full nonlinear 
	upward cascade of turbulent energy 
	for high-rotation cases.

\item The formation of the easterly circumpolar vortex is a robust
      feature of the two dimensional decaying turbulence on a rotating
      sphere and insensitive to the choice of the initial energy
      spectrum. 

\item The zonal band structures are discernible from an early stage of
      the time evolution, and once established their latitudinal
      positions are hardly changed with time.  The number of the jets
      increases while the width of them decreases as the rotation rate
      increases. 

\item Two dimensional 
	energy spectrum shows
	anisotropy due to the rotation effect of the sphere. 
	An airfoil shape with very small energy density appears 
	in low total-wavenumber region.  
	Anisotropic distribution of the energy
	is also found in the high total-wavenumber region.

\end{enumerate}

\vspace{1cm}
\begin{center}
{* \ * \ *}
\end{center}

The GFD-DENNOU Library (SGKS Group 1995) 
was used for drawing the figures. 
Numerical calculation was done on VPP500/15 
	at the Data Processing Center, Kyoto University. 
This work was supported in part by the Grant-in-Aid for
Scientific Research of the Ministry of Education, Science, Sports and
Culture of Japan and by the Grant-in-Aid for the Research for the
Future Program ``Computational Science and Engineering'' of the Japan
Society for the Promotion of Science.

\appendix 

\section*{REFERENCES}
\begin{description}

\item SGKS Group, 1995: 
	DCL-5.1. 
	http://www.gfd-dennou.org/library/dcl/\~, 
	GFD-Dennou Club (in Japanese).

\item Hayashi, Y.-Y. and Matsuno, T., 1984:
	Amplitude of Rossby wavetrains on a sphere.  
	{\it J. Met. Soc. Japan,} {\bf 62}, 377-387.

\item Hoskins, B.J. and Karoly, D.J., 1981:
	The steady linear response of a spherical atmosphere 
	to thermal and orographic forcing.
	{\it J. Atmos. Sci.}, {\bf 38}, 1179-1196.

\item Nozawa, T. and Yoden, S., 1997a: 
	Formation of zonal band structure in forced two-dimensional
	turbulence on a rotating sphere. 
	{\it Phys. Fluids}, {\bf 9}, 2081-2093.

\item Nozawa, T. and Yoden, S., 1997b: 
	Spectral anisotropy in forced two-dimensional turbulence
	on a rotating sphere. 
	{\it Phys. Fluids}, {\bf 9}, 3834-3842.

\item Rhines, P.B., 1975: 
	Waves and turbulence on a beta-plane.
	{\it J. Fluid Mech.}, {\bf 69}, 417-443.

\item Rhines, P.B., 1994: 
	Jets.
	{\it Chaos}, {\bf 4}, 313-339.

\item Vallis, G.K. and Maltrud, M.E., 1993:
	Generation of mean flows and jets on a beta plane and 
	over topography. 
	{\it J. Phys. Oceanogr.}, {\bf 23}, 1346-1362.

\item Williams, G.P., 1978: 
	Planetary circulations: 
	1. Barotropic representation of Jovian and Terrestrial turbulence.
	{\it J. Atmos. Sci.}, {\bf 35}, 1399-1426.

\item Yoden, S. and Yamada, M., 1993:
	A numerical experiment on two-dimensional decaying turbulence 
	on a rotating sphere. 
	{\it J. Atmos. Sci.}, {\bf 50}, 631.

\end{description}

\end{document}





