\documentstyle[proceedings,numreferences]{crckapb}
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\begin{opening}
\title{
  LARGE-SCALE KOLMOGOROV FLOW ON THE BETA-PLANE,
  RESONANT WAVE INTERACTIONS AND SCALE SELECTION
}
\runningtitle{LARGE-SCALE KOLMOGOROV FLOW ON THE BETA-PLANE}

\author{U. FRISCH$^1$, B. LEGRAS$^2$ AND B. VILLONE$^3$}
\institute{$^1$ CNRS, Observatoire de Nice, BP 4229, 06304 Nice Cedex 4, 
France.\\
$^2$ CNRS/LMD/ENS, 24 rue Lhomond, 75231 Paris Cedex 5, France.\\
$^3$ Istituto di Cosmogeofisica, CNR, C. Fiume 4, 10133 Torino, Italy}
\end{opening}
\runningauthor{U. FRISCH, B. LEGRAS AND B. VILLONE}
\begin{document}
The large-scale dynamics of the Kolmogorov flow near its threshold of
instability is studied in the presence of the $\beta$-effect (Rossby
waves). The governing equation, obtained by a multiscale technique,
fails the Painlev\'e test of integrability when $\beta\ne0$. This
``$\beta$-Cahn--Hilliard'' equation with cubic nonlinearity is
simulated numerically in various r\'egimes. The dispersive action of
the waves modifies the inverse cascade associated with the Kolmogorov
flow \cite{she}. For small values of $\beta$ the inverse cascade is
interrupted at a wavenumber which increases with $\beta$.  For large
values of $\beta$ only resonant wave interactions (RWI) survive.  An
original approach to RWI is developed, based on a reduction to normal
form, of the sort used in celestial mechanics \cite{Arno}.  Otherwise,
wavenumber discreteness effects, which are dramatic in the present
case, are not captured. The method is extendable to arbitrary RWI
problems of the kind encountered in plasma physics, spin waves,
oceanography, etc. (see, e.g., Ref.~\cite{ZLF}). The only four-wave
resonances present involve two pairs of opposite wavenumbers. This
allows leading-order decoupling of moduli and phases of the various
Fourier modes, so that an exact kinetic equation is obtained for the
energies of the modes.  It has a Lyapunov functional (gradient)
formulation and multiple attracting steady-states, each with a single
mode excited. The final state depends thus on the initial condition
chosen, as illustrated in Fig.~1.  A detailed presentation of this
work may be found in Ref.~\cite{flvpd}. 


Recent calculations, suggested by V.~Yakhot, have been performed with
a large number $n$ of linearly unstable modes and a small value of
$\beta$, such that the $\beta$-effect is important only up to some
wavenumber $1<k_\beta \ll n$, thereby selecting a preferred scale
$\sim k_{\beta}^{-1}$. After relaxation of transients the solution
becomes a slowly traveling wave; the only wavenumbers excited are
then (odd) multiples of $k_\beta$\,. The energies of the modes are
very close to those obtained for {\em unstable\/} steady-state
solutions of the Cahn--Hilliard equation with $k_\beta$ pairs of
kinks/antikinks
\cite{KO82}, which are thus stabilized by the combined effect of the
Rossby waves and the slow drift.

\begin{figure}
\centerline{\psfig{file=etc-example-fig.ps,width=7.5cm,clip=}}
\caption{Simulation of the $\beta$-Cahn--Hilliard equation
with strong Rossby waves when the dynamics are dominated by resonant
wave interactions.  At long times the Fourier amplitudes go to a
steady state with a single Fourier mode excited, as predicted by the
asymptotic theory.  Several single--Fourier
mode attractors are competing, as indicated by (a) and (b) which
correspond to two slightly different initial conditions.}
\end{figure}

\begin{thebibliography}{99}

\bibitem{she} She, Z.S. (1987) Metastability and vortex
pairing in the Kolmogorov flow, {\it Phys. Lett.} A{\bf 124},
pp.~161--164.

\bibitem{Arno} Arnold, V.I., Kozlov, V.V. \& Neishtadt, A.
(1988) Mathematical aspects of classical and celestial mechanics, 
in {\it Dynamical Systems III}, ed.~V.I.~Arnold,  pp.~1--291,
Encyclopaedia of Mathematical Sciences, vol.~3, Springer Verlag.

\bibitem{ZLF} Zakharov, V.E., L'vov, V.S. \& Falkovich, G.
(1992) {\it Kolmogorov Spectra of Turbulence I\/}, Springer
Verlag.

\bibitem{flvpd} Frisch, U., Legras, B. and Villone, B. (1996)
Large-scale Kolmogorov flow on the beta-plane and resonant wave
interactions, {\it Physica D}, in press.

\bibitem{KO82} Kawasaki, K. \& Ohta, T. (1982) Kink dynamics
in one-dimensional nonlinear systems, {\it Physica} A{\bf
116}, 573--593.

\end{thebibliography}


\end{document}