\documentclass{elsart}
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\begin{document} 

\begin{frontmatter}

\title{DOMAIN GROWTH AND COARSENING INHIBITION IN A NON POTENTIAL SYSTEM}

\author{R. Gallego, M. San Miguel and R. Toral} 
\address{Instituto Mediterr\'{a}neo de Estudios Avanzados, IMEDEA (CSIC-UIB)\\ 
Campus Universitat Illes Balears, E-07071 Palma de Mallorca, Spain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ABSTRACT%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract} 

We present a study of interface dynamics in two spatial dimensions for a
non-relaxational system that describes the temporal evolution of three
competing real fields. This and similar models have been used to get insight
into problems like Rayleigh-B\'enard convection in a rotating cell or
population competition dynamics in predator-key systems. A notable feature is
that the non-potential dynamics stops the coarsening process as long as the
system size is large enough. For certain values of the parameters, the system
switches to a chaotic dynamical state known as the K\"uppers-Lortz (KL)
instability. When isotropic spatial derivatives are used, the intrinsic period
of the KL instability diverges with time. On the contrary, anisotropic
derivatives stabilize the KL period.

\end{abstract} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frontmatter}


%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and model}
%%%%%%%%%%%%%%%%%%%%%%

A topic that has deserved recently much attention is the influence of
nonpotential dynamics (those that cannot be obtained from the minimization of a
potential function) on coarsening processes. In this paper we show how, for
nonpotential systems, different types of spatial derivatives can modify
completely the nature of the coarsening process. For this purpose, we use
a theoretical model proposed in the context of convection in a rotating cell by
Busse and Heikes~\cite{Busse} to which spatial dependent terms have been added:

\begin{equation}\label{eq:modelo}
  \begin{array}{rcl}
  \partial_{t} A_{1} &=& \mathcal{L}_1A_{1}+A_{1}\, [1-A_{1}^{2}
  -(\eta + \delta)\, A_{2}^{2}-(\eta-\delta)\, A_{3}^{2}],	\\
  \partial_{t} A_{2} &=& \mathcal{L}_2A_{2}+A_{2}\, [1-A_{2}^{2}
  -(\eta + \delta)\, A_{3}^{2}-(\eta-\delta)\, A_{1}^{2}],	\\
  \partial_{t} A_{3} &=& \mathcal{L}_3A_{3}+A_{3}\, [1-A_{3}^{2}
  -(\eta + \delta)\, A_{1}^{2}-(\eta-\delta)\, A_{2}^{2}].
  \end{array}
\end{equation}

\noindent Here $\mathcal{L}_i\ (i=1,2,3)$ are linear differential operators.
The fields $A_i\ (i=1,2,3)$ are the (real) amplitudes of three set of
convection rolls oriented 60 degrees at each other; the parameter $\delta$
bears on the rotation angular velocity of the cell and $\eta$ is related to
other physical properties of the fluid. Similar models have been used to study
competition population dynamics in predator-prey
systems~\cite{Frachebourg,May}. A one-dimensional version of (\ref{eq:modelo})
is studied in reference~\cite{GallegoPRE}. 

It is convenient to split (\ref{eq:modelo}) into potential and non-potential
contributions: $\partial_tA_i=\delta\mathcal{F}/\delta A_i+\delta\cdot f_i\
(i=1,2,3)$, where $\mathcal{F}$ is a real functional. When $\delta=0$,
$\mathcal{F}$ is a Lyapunov potential, so that the dynamics relaxes towards the
minima of $\mathcal{F}$. On the other hand, $\delta\neq0$ implies a
non-relaxational dynamics governing the system. In the fluid analogy this means
a nonzero rotation angular velocity.

Eqs. (\ref{eq:modelo}) have three stationary homogeneous ``roll'' solutions
$A_i=1$, $A_j=0$, $j\neq i=1,2,3$ that are stable for $|\delta|<\eta-1$. For
every $\eta>1$ there exists a critical value $\delta_c(\eta)$ such that, when
$|\delta|>\delta_c$, the roll solutions are no longer stable. Then the system
breaks up into a chaotic dynamical state. In the reference system that rotates
with the cell, the convective rolls alternatively change between three
preferred directions. This phenomenon is known as the K\"uppers-Lortz (KL)
instability. The $0$-dimensional model ($\mathcal{L}_i=0,\ i=1,2,3$) shows the
unwanted feature that the period between successive alternations of the
dominating modes $A_1$, $A_2$ and $A_3$ diverges with time. In order to avoid
this behavior, two solutions have been proposed. Originally, Busse and Heikes
suggested that noise (present at all times) could lead to a constant period
(fluctuating around a mean value). This is checked in numerical
simulations~\cite{RaulReport}. Another solution to circumvent this problem is
to add spatial dependent terms to the equations. Starting from a small
perturbation of the unstable state $A_1=A_2=A_3=0$ in the case $\eta>1$, the
system develops a spatial structure consisting of domains of rolls separated by
rather abrupt interfaces. In the bulk of each domain, one amplitude is close to
one and the other two close to zero. The KL instability now occurs in the bulk
of the domains and leads to an alternation of the modes at every point of
space. Below the KL point the system may exhibit domain growth until it reaches
a final stationary state.

%This phenomenon
%is known as the K\"uppers-Lortz (KL) instability. It occurs in the bulk of the
%domains and leads to an alternation of the modes at every point of space. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The role of the spatial derivatives}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We use two choices for the linear differential operators $\mathcal{L}_i$
of model~(\ref{eq:modelo}):

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Isotropic derivatives}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

First we consider terms of the simplest diffusive form, i.e.,
$\mathcal{L}_i=\nabla^2$, $i=1,2,3$. This kind of spatial dependent terms have
also been used in biological models~\cite{Frachebourg}.

The isotropic nature of the spatial dependent terms make the fronts move in the
normal direction at each point. It is possible to show~\cite{unpublished} that
the (normal) front velocity is given by
$v_n(\mathbf{r},t;\eta,\delta)=-\kappa(\mathbf{r},t)+\delta\cdot f(\eta)$,
where $\kappa$ is the local curvature of the front line. The term
$v(\delta)\equiv\delta\cdot f(\eta)$ is the planar front velocity which is
proportional to $\delta$ at lowest order. For $\eta,\delta$ fixed, there exists
a critical value of the curvature, $\kappa_c\equiv\delta\cdot f(\eta)$ such
that an interface does not propagate. In spherical symmetry this fact entails
the existence of a droplet of critical radius $R_c$; any drop with radius
$R>R_c$ grows and if $R<R_c$ it shrinks.

In the potential limit, $\delta=0$, the system tends to reduce the total
interfacial area by decreasing the curvature of interfaces. As a consequence,
domains grow in time and the system shows coarsening. The growth law is found
to be $R(t)\sim t^{1/2}$ as corresponds to a nonconserved scalar order
parameter. However when $\delta\neq 0$, the nonpotential dynamics can stop the
coarsening process even if $\delta<\delta_c$.   The role of the non-potential
terms is mainly to make the front lines rotate around points (vertices) where
the three amplitudes coexist and this prevents the system from coarsening for
system sizes sufficiently large.  It is important to realize that the
alternation between the modes in a fixed point above the KL instability is the
result of two combined phenomena: the KL instability itself in the bulk of the
domains and the interface rotation around the vertices. For relative small
systems, the vertices are close to each other and those with opposite sense of
rotation may annihilate each other leading to coarsening. In a noncoarsening
situation, for a large enough system, the number of vertices is found to be
constant for long times. The vertices interact each other and move through the
system with a time scale much larger than that associated with interface
motion. The two images of the upper row in figure \ref{fig} show two
configurations in two dimensions at long times below and beyond the instability
point. Apart from the typical size of the domains, it appears that there is no
qualitative difference between them. The reason is that, just like in the zero
dimensional case, the KL period $T$ increases with time. Therefore, at long
times,  $T$ is so large that we only see rotating interfaces around vertices as
below the instability point.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Anisotropic derivatives}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The second choice for the spatial derivatives is a simplified version of the
Newell-Whitehead-Segel type terms retaining dominant contributions as made
in~\cite{Tu}. Now $\mathcal{L}_i=\partial^2_{\hat{x}_i}$, $i=1,2,3$, where
$\hat{x}_1,\hat{x}_2,\hat{x}_3$ represent three directions with a relative
orientation of $60^\circ$. The anisotropic nature of these spatial terms
changes the dynamics significantly with respect to the isotropic derivatives.
In particular the propagation of the interfaces no longer follows the normal
direction at each point. A consequence is that closed domains adopt an elliptic
shape rather than a spherical one. Moreover the number of vertices is not
constant for long times but they annihilate each other and also originate from
the collision of interfaces (two interfaces rotating in the same direction may
collide and generate new vertices).

The returning time of the KL instability now saturates to a finite value so
directional derivatives seem to be more indicated to study the KL instability
than isotropic ones. However it is more difficult to make analytical studies of
interface dynamics.

The two images on the lower row of figure \ref{fig} represent two typical
configurations below and beyond the KL instability with directional spatial
derivatives terms. Below the KL instability we observe rotating interfaces that
inhibit coarsening as in the case with isotropic derivatives. In the KL regime
we observe in addition domains of one phase emerging in the bulk of other
domain.

\begin{thebibliography}{1}

\bibitem{Busse}
F.~H. Busse and K.~E. Heikes, {\em Science} {\bfseries 208}, 173 (1980).

\bibitem{Frachebourg}
L.~Frachebourg, P.~L. Krapivsky, and E.~Ben-Naim, {\em Phys. Rev. E} 
{\bfseries 54}, 6186 (1996).

\bibitem{Tu}
Y.~Tu and M.~C. Cross, {\em Phys. Rev. Lett.} {\bfseries 69}, 2515 (1992).

\bibitem{May}
R. M. May and W. J. Leonard, {\em SIAM J. Appl. Math} {\bfseries 29}, 
243 (1975).

\bibitem{GallegoPRE}
R. Gallego, M. San Miguel, and R. Toral, {\em Phys. Rev. E} {\bfseries 58}, 
3125 (1998).

\bibitem{RaulReport}
M. San Miguel and R. Toral, in \emph{Stochastic Effects in Physical Systems}, 
Instabilities and Nonequilibrium Structures VI, Ed. E. Tirapegui, 
Kluwer Academic, (1998).

\bibitem{unpublished}
R. Gallego, M. San Miguel, and R. Toral, unpublished.

\bibitem{Montagne}
R.~Montagne, E.~Hern{\'a}ndez-Garc{\'{\i}}a, A. Amengual and M.~San Miguel,
{\em Phys. Rev. E} {\bfseries 96}, 151 (1997).

\end{thebibliography}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIGURES%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[h]
\begin{center}
  \epsfig{file=/home/rafa/figures/postscript/figCCP98_2.eps, width=0.7\textwidth}  
  \caption{Snapshots corresponding to a numerical simulation in two dimensions
  of the system (\ref{eq:modelo}) for both isotropic and directional
  derivatives below and beyond the KL instability. The white, grey and black
  regions are the regions occupied by the modes $A_1$, $A_2$ and $A_3$
  respectively. To integrate the system of equations~(\ref{eq:modelo}) we have
  used a finite differential scheme in a rectangular mesh with null boundary
  conditions.} \label{fig}
\end{center}
\end{figure}


\end{document}