%\documentstyle[aps,eqsecnum,epsf]{revtex} \documentstyle[aps,twocolumn,eqsecnum,epsf]{revtex} \tightenlines %\documentstyle[11pt]{article} %\documentstyle[11pt,ami]{article} %\renewcommand{\baselinestretch}{1.5} %\renewcommand{\baselinestretch}{1.} %\setlength{\parindent}{0pt} \begin{document} \title{Pattern Selection and the Effect of Group Velocity on Interacting Oscillatory and Stationary Instabilities} \author{D.Walgraef \cite{Daniel}} \address{Departament de F{\'\i}sica, Universitat de les Illes Balears,\\ E-07071 Palma de Mallorca, Spain} \date{\today} \maketitle \begin{abstract} The effect of mean flows on pattern stability in systems where oscillatory instabilities of the Hopf type interact with stationary ones is investigated. In particular, it is shown that pattern selection may be strongly modified when the absolute instability threshold of the trivial uniform steady state is rejected beyond the stationary instability. The effect of spatially distributed noise including the competition between noise sustained and dynamically sustained structures is also discussed.\\ \\ PACS: 47.20.Ky, 05.40.+j, 43.50+y, 47.50+d \end{abstract} \section{Introduction} Several physico-chemical systems driven out of equilibrium present oscillatory instabilities or Hopf bifurcations leading to the formation of spatio-temporal wave patterns. Celebrated examples are Rayleigh-B\'enard instabilities in binary fluids \cite{binary}, electrohydrodynamic instabilities in nematic liquid crystals \cite{ehd}, or convective instabilities in Taylor-Couette systems \cite{taylorcouette}. Close to the instability, the systems dynamics may be reduced, in one-dimensional geometries, to coupled complex Ginzburg-Landau equations which describe the evolution of the amplitude of counter-propagating waves that may appear beyond the bifurcation point \cite{crosshoh}. The coefficients of these equations have been evaluated by means of analytical and numerical techniques, for binary fluid convection for different separation ratios, Prandtl numbers and Lewis numbers \cite{zimmer}, and for polymeric fluid convection, for different fluid characteristics\cite{ucv}. From the values of these kinetic coefficients of these equations, which have been derived directly from the Navier-Stokes equations, it appears that, in binary fluid convection, the selected pattern should correspond to traveling waves, while in visco-elastic convection, there is a wide range of parameters where the selected stable patterns should correspond to standing waves. Furthermore, in the latter case, the amplitude and phase stability of these standing waves versus extended perturbations have been computed for a series of typical values of the parameters corresponding to polymeric solutions ranging from Jeffreys to maxwellian ones \cite{ucv1}. \medskip However, these Ginzburg-Landau equations contain mean flow terms induced by group velocities whose importance varies according to the fluid under consideration. As a result, one also has to study the convective and absolute stability of the wave patterns. Let us recall that, when the reference state is convectively unstable, localized perturbations are driven by the mean flow in such a way that they grow in the moving reference frame, but decay at any fixed location. On the contrary, in the absolute instability regime, localized perturbations grow at any fixed location \cite{huerre}. As a result, the behavior of the system is qualitatively very different in both regimes. In the convectively unstable regime, a deterministic system cannot develop the expected wave patterns, except in particular experimental set-ups (e.g. in annular containers or in the presence of reflecting boundaries), while in a stochastic system, noise is spatially amplified and gives rise to noise-sustained structures \cite{deissler}. On the contrary, in the absolutely unstable regime, waves are intrinsically sustained by the deterministic dynamics, which provides the relevant selection and stability criteria \cite{mueller,buechel}. Hence, the concepts of convective and absolute instability are essential to understand the behavior of nonlinear wave patterns and their stability \cite{deissler,mutveit}. \medskip Noise sustained wave patterns have been widely studied, first, in systems where single or counter-propagating traveling waves are preferred, and, more recently, in systems where it is standing waves that are the preferred structures \cite{deissler,deisslerbrand,helmutda,ahlers,steinberg,luecke,marc}. In particular, in the latter case, it has been shown that, in deterministic systems transitions from the conduction state to traveling waves, and, finally, to standing waves occur at thresholds that depend on the group velocity, while in stochastic systems, standing waves are sustained by noise in all the parameter range beyond the Hopf bifurcation \cite{marc}. \medskip Besides the oscillatory instabilities, many of these systems also present stationary instabilities leading to steady spatial patterns. Which type of convection appears first, oscillatory or stationary, is determined, in polymeric fluids, by their rheological parameters. In particular, at fixed Prandtl numbers, it is the stress relaxation time that fixes the relative position of each instability threshold. Hence, in the case where oscillatory instability appears first, the corresponding absolute instability may nevertheless be rejected beyond the stationary instability, if the group velocity is sufficiently large, such as in Maxwell fluids. In this case, the difference between deterministic and stochastic systems should be qualitative in nature. Effectively, in deterministic systems, stationary patterns should develop first, even though the Hopf bifurcation is the first to appear, while in stochastic systems, standing waves should be sustained by noise in beyond the Hopf bifurcation. \medskip To describe this situation, one needs to consider coupled amplitude equations for interacting oscillatory and steady modes. The aim of this paper is to achieve a qualitative understanding of the behavior of such dynamical systems for arbitrary values of the cross-coupling coefficients and of the group velocity, and to analyze the effect of these parameters on the pattern selection, either in deterministic and stochastic systems. \medskip The paper is organized as follows. In section \ref{vcgle}, amplitude equations for interacting oscillatory and stationary modes are presented in the general case. In section \ref{instability}, convective and absolute instability thresholds are determined for the different ground states when the cross coupling between counter-propagating waves sustains traveling waves or standing waves. The resulting pattern selection is analyzed for deterministic systems in section \ref{detpatsel} and for stochastic ones in section \ref{stochpatsel}. Numerical checks are presented in section \ref{numerical} and conclusions are drawn in section \ref{conclusion}. \medskip \section{Amplitude equations for interacting oscillatory and stationary instabilities}\label{vcgle} \medskip Let us consider a driven physico-chemical system where an oscillatory instability, say a Hopf bifurcation with broken spatial symmetry, and a stationary pattern forming instability are close together. In their vicinity, the order parameter-like variable may be written, in one-dimensional horizontal geometries, as : \begin{eqnarray}\vec u(\vec r,t) &=& [\vec u_0(z)(A(X,T) e^{i(k_cx +\omega_ct)} + B(X,T)e^{-i(k_cx -\omega_ct)} \nonumber\\&+& R(X,T) e^{ik_0x} ) + c.c.] \end{eqnarray} where $k_c$ and $k_0$ are the critical wavenumber associated with each instability. The amplitudes, $A(X,T)$, $B(X,T)$ and $R(X,T)$, depend on the slow variables $X=\epsilon^{1/2}x$ and $T=\epsilon^{-1}t$, where $\epsilon$ is the reduced distance to one of the instability thresholds, say the Hopf instability threshold ($\epsilon = {B-B_h \over B_h}$ where B is the bifurcation parameter and $B_h$ its critical value at the Hopf bifurcation). The structure of their evolution equations may easily be obtained on using the symmetries of the problem and correspond to the following coupled Ginzburg-Landau equations : \begin{eqnarray}\label{CGL2} \partial_T A &=& V \partial_X A + \epsilon A + (1+i\alpha) \partial_X^2 A - (1+i \beta)\vert A\vert^2 A \nonumber \\ &-& \gamma(1+i \delta)\vert B\vert^2 A - u(1+i v)\vert R\vert^2 A \nonumber \\ \partial_T B &=&- V\partial_X B +\epsilon B + (1+i\alpha)\partial_X^2 B - (1+i \beta) \vert B\vert^2 B \nonumber \\&-& \gamma(1+i\delta)\vert A\vert^2 B - u(1+i v) \vert R\vert^2 B \nonumber \\ \tau \partial_T R&=& (\epsilon -\epsilon_s )R + \xi_0^2 \partial_X^2 R - \vert R\vert^2 R \nonumber \\&-& w(1+i\zeta)(\vert A\vert^2 +\vert B\vert^2) R \end{eqnarray} where $\epsilon_s = (B_s-B_h)/B_h$ is positive when the oscillatory instability is the first to appear on increasing the bifurcation parameter, which is the case we will consider here. $\xi_0^2$ is related to dispersive effects; $\gamma$ and $\delta$ are the cross-coupling coefficients between oscillatory modes (I will consider $-1<\gamma$, in order to ensure supercritical bifurcations); u, $\nu$, w and $\zeta$ are the cross-coupling coefficients between oscillatory and steady modes. The evolution equations for the amplitudes A and B in the absence of interactions with stationary modes have been derived and studied in different contexts \cite{crosshoh,CoulletFauve,Fauve,bes1,deisslerbrand,CoulletFrisch}, and it is now well known that it is the non-linear cross-coupling term between both amplitudes that determines if the stable patterns correspond to traveling (strong cross-coupling) or standing (weak cross-coupling) waves. Their generalization to the co-dimension two problem has been studied in the framework of Rayleigh-Benard convection in binary fluids \cite{brandco2,zimmer} where the bifurcation parameter is the Rayleigh number and where the wave cross-coupling term $\gamma$ is larger than one and favors traveling waves. Up to now, they have not been studied for low wave cross-couplings ($\gamma < 1$) which favor standing waves. This is the case of viscoelastic convection where the amplitude equations have been calculated for non-interacting oscillatory and stationary instabilities only \cite{ucv}. Hence, the coefficients $u(1+i \nu)$ and $w(1+i\zeta)$ are not known yet. I will nevertheless study these equations for $\gamma < 1$ and arbitrary cross-coupling coefficients $u(1+i \nu)$ and $w(1+i\zeta)$ in order to assess the possible behavior of the solutions of these equations. The application of the results to viscoelastic convection should improve the qualitative understanding of pattern formation in polymeric solutions, and provide useful hints for the interpretation of experimental results \cite{kolodner}. \section{Stability of the ground states}\label{instability} In order to determine the patterns that may be selected by the dynamics (\ref{CGL2}), one has to analyze the stability of the different steady state solutions, and, in the first place, of the trivial conduction state. Since eq. (\ref{CGL2}) correspond to supercritical bifurcations, thus with stabilizing non linearities, this study may be performed through a linear stability analysis \subsection{Stability of the conduction state} On linearizing the equations (\ref{CGL2}) around the trivial solution $A(X,T)=B(X,T)=R(X,T)=0$ , the complex growth rates of disturbances of wavenumber $\kappa$ satisfy the dispersion relations : \begin{eqnarray} \omega_A &=& KV+ [\epsilon +(1+i\alpha)K^2] \nonumber\\ \omega_B &=& - KV+ [\epsilon +(1+i\alpha)K^2] \quad,\quad K=k+iq\quad \nonumber\\ \omega_R &=& {1\over \tau}[(\epsilon -\epsilon_s )+\xi_0^2K^2] \end{eqnarray} and the growth rates of such perturbations are given by $\Re \omega (K)$. Using the method of steepest descent, the long-time behavior of the system along a ray defined by fixed x/t, i.e. in a frame moving with a velocity $v_0 = x/t$, is governed by the saddle point defined by : \begin{equation}\label{reimv}\Re\left(\frac{d\omega}{dK}\right)\vert_{K_0}= 0 \quad , \quad \Im \left(\frac{d\omega}{dK}\right)\vert_{K_0}=v_0 \end{equation} Since absolute instability occurs when perturbations grow at fixed locations, one has to consider the growth rate of modes evolving with zero group velocity, which are defined by : \begin{equation} \label{reim} \Re \left(\frac{d\omega}{dK}\right)= \Im\left(\frac{d\omega}{dK}\right)=0 \end{equation} These conditions define the following wave number \begin{eqnarray}\label{wavevekt} q_{A(B)}&=&-\alpha k_{A(B)}\quad , \quad q_R=k_R=0\\ k_{A(B)} &=& \mp\frac{V}{2(1+\alpha^2)}\quad.\nonumber \end{eqnarray} The real part of $\omega$, which determines the growth rate $\lambda$ of these modes is then : \begin{eqnarray} \lambda_{A(B)}&=&\Re(\omega_{A(B)}) =\epsilon-\frac{V^2} {4(1+ \alpha^2)} \nonumber \\ \lambda_R&=&\Re(\omega_R) =\epsilon - \epsilon_s \quad .\end{eqnarray} Therefore, the trivial conduction state is absolutely unstable if $\lambda>0$. As already shown in \cite{deissler}, this condition determines a critical line in the parameter space which can be expressed for the group velocity $V$ or the control parameter $\epsilon$ as \begin{equation} v_c=2\sqrt{\epsilon (1+\alpha^2)} \qquad \mbox{or}\qquad \epsilon_c=\frac{V^2}{4(1+\alpha^2)}. \end{equation} Hence, for $0< \epsilon < \epsilon_c$, the conduction state is convectively unstable towards wavy modes, and wave patterns are unable to develop in the absence of noise. For $\epsilon >\epsilon_c$, wave patterns may grow and are sustained be the dynamics, even in the absence of noise. On the other hand, for $0< \epsilon < \epsilon_s $ the conduction state is stable versus stationary modes, and unstable for $\epsilon_s < \epsilon$ . Hence, for $\epsilon >\epsilon_c$ and $\epsilon >\epsilon_s $ both types of modes may start growing, but it is of course their non-linear interactions that will determine the resulting patterns and their stability. Let me then consider the different possibilities which are pure wave patterns, pure roll patterns and mixed states involving rolls and wave patterns. I will consider here uniform amplitude solutions corresponding to spatio-temporal patterns with critical wavenumbers. The stability of modulated or wave solutions will be considered later on. \subsection{Stability of pure wave patterns} \paragraph{Traveling waves} A first class of non trivial steady states of the dynamical system (\ref{CGL2}) corresponds pure critical traveling waves solutions $A(X,T)_0=\sqrt\epsilon \exp{-i(\beta\epsilon T +\phi)}$, $B(X,T)=R(X,T)=0$, or $B_0(X,T)=\sqrt\epsilon \exp{-i(\beta\epsilon T +\phi)}$, $A(X,T)=R(X,T)=0$, where $\phi$ is an arbitrary phase. One may consider the first family without loss of generality, and, in order to study its linear stability, one has to look for solutions in the form $|A|=(\sqrt{\mu } + a) exp -i\beta\mu t,\quad B(x,t)=b$, and compute the eigenvalues of the linearized evolution equations for $a$, $b$ and heir complex conjugates. The real parts of the eigenvalues of the Fourier transform of $a$ are well known (see for example \cite{crosshoh}) and read : \begin{eqnarray} \Re \omega_{\vert a\vert}&=&-2\mu - (1-\alpha\beta)q^2 + ...\nonumber \\ \Re \omega_{\phi}&=& - (1+\alpha\beta)q^2 - {\alpha^2(1+\beta^2)\over2\mu}q^4 +... \end{eqnarray} The first one is always negative, but the second one may become positive and the system may experience a Benjamin-Feir instability when $1+\alpha\beta $ is negative \cite{BF,newell}. In the following, we will consider systems where $\alpha$ and $\beta$ are sufficiently small and positive, such that $1+\alpha\beta >0$ . The only remaining instability mechanism may then result from the growth of B. Effectively, the linearized evolution equations for $b$ or $\bar b$ give the following growth rate : \begin{equation} \omega_B=\mu(1-\gamma) -Kv+(1+i\alpha)K^2 . \end{equation} As in the preceding section, the conditions (\ref{reim},\ref{wavevekt}) determine $K$. The stability of the solution ($\sqrt{\epsilon} exp-i(\beta\epsilon T +\phi), 0, 0$) is determined by the growth rates $\lambda_j=\Re(\omega_j)$ ( $j=A,B,R$). Since $\omega_A$ is negative, we get the following condition for absolute stability of a the pure traveling wave states \begin{eqnarray} \lambda_B &=&\Re(\omega_B)=\epsilon(1-\gamma)-\frac{V^2} {4(1+\alpha^2)}< 0\quad , \nonumber\\ \lambda_R&=&\Re(\omega_R)=\epsilon(1-w)-\epsilon_s < 0\quad , \end{eqnarray} Hence, for $\gamma >1$ and $w>1$, pure traveling waves are stable, while they may become unstable for $\gamma <1$ and/or $w<1$. \medskip For $w<1$, traveling waves are stable for $0<\epsilon <{\epsilon_s \over 1-w}$ and become unstable versus spatial modulations for $\epsilon >{\epsilon_s \over 1-w}$. For $-1<\gamma <1$, one has to distinguish between the following cases : \medskip (1) $w > 1$ : stationary spatial modulations decay and pure traveling waves are thus convectively unstable, but absolutely stable versus counterpropagating wavy modes for $0< \epsilon < \epsilon_c'= \epsilon_c/(1-\gamma)$, and absolutely unstable for $\epsilon_c'<\epsilon$. The corresponding critical group velocity is $v_c'=v_c\sqrt{1-\gamma}$. As a result, on increasing the bifurcation parameter in deterministic systems with $0<\gamma <1$, traveling waves should be expected for $\epsilon_c<\epsilon < \epsilon_c'$, as shown in ref.\cite{marc}. However, when $\gamma <0$, which is the case in viscoelastic convection \cite{ucv}, $\epsilon _c'<\epsilon _c$. Hence, in deterministic dynamics, traveling wave state cannot be obtained, in ramp experiments, from the trivial conduction state. \medskip (2) $w < 1$ : the absolute and convective stability properties of pure traveling waves versus counterpropagating wavy modes remain unchanged, but they are unstable versus stationary spatial modulations for $\epsilon > {\epsilon_s \over 1-w}$. Pure traveling waves are thus only convectively unstable for $\epsilon < min(\epsilon_c', {\epsilon_s \over 1-w})$. Hence, for $0<\gamma <1$, they may only be expected when $\epsilon_c < {\epsilon_s \over 1-w}$ for $\epsilon_c< \epsilon < min(\epsilon_c', {\epsilon_s \over 1-w})$ while for $\gamma <0$, they still cannot be obtained from the trivial conduction state. \paragraph{Stability of standing waves} A second class of non trivial steady states of the dynamical system (\ref{CGL2}) corresponds to the pure critical standing waves solutions $A_s(X,T)=\sqrt{\epsilon \over 1+\gamma}exp-i({\beta +\gamma\delta \over 1+\gamma}\epsilon T +\phi)$, $B_s(X,T)=\sqrt{\epsilon \over 1+\gamma} exp-i({\beta +\gamma\delta \over 1+\gamma}\epsilon T +\psi)$, $R(X,T)=0$, where $\phi$ ans $\psi$ are arbitrary phases. \medskip For $\gamma >1$, standing waves are known to be unstable. \medskip For $\gamma <1$, standing waves are stable versus perturbations in A and B, provided $1+\alpha{\beta -\gamma^2\delta \over 1-\gamma^2}>0$ \cite{CoulletFauve} (which reduces to the habitual Benjamin-Feir criterion $1+\alpha\beta >0$ in the special case where $\delta =\beta$ \cite{maxi}). I will consider here that these conditions are satisfied, as it is usually the case for convection in Oldroyd-B viscoelastic fluids \cite{ucv1}, which is a typical example of system where $\gamma <1$. Note that the results obtained below are also valid for non critical standing wave solutions in their phase stability domain (the phase stability condition being derived in \cite{ucv1}). Outside this domain, one needs to study the convective-absolute stability of the corresponding patterns, an aspect that will be analyzed later on. For the time being, let us only consider critical standing waves which are stable versus wavy mode perturbations. \medskip The full stability analysis also requires to study the growth rate of spatial disturbances in R around the state ($A_s$, $B_s$,0), which is given by: \begin{equation} \lambda_R=\Re(\omega_R)=\epsilon(1-{2w\over 1+\gamma})-\epsilon_s \end{equation} In these conditions, critical standing waves are thus stable for any $\epsilon >0$ if $2w > 1+\gamma $ , and for $0<\epsilon < \epsilon_s {1+ \gamma \over 1+\gamma -2w}$ if $2w < 1+\gamma $. \subsection{Stability of steady rolls} The linear growth rates of the wavy mode perturbations around the pure critical roll state (0,0, $R_0=\sqrt{\epsilon -\epsilon_s }expi\phi$) which may exist for $\epsilon -\epsilon_s $ are given by : \begin{eqnarray} \omega_A &=& KV+ [\epsilon +(1+i\alpha)K^2] - u(1+iv)(\epsilon -\epsilon_s ) \nonumber\\ \omega_B &=& - KV+ [\epsilon +(1+i\alpha)K^2] - u(1+iv)(\epsilon -\epsilon_s ) \quad , \nonumber \\\ K&=&k+iq\quad .\end{eqnarray} The standard analysis shows that this state is stable for : \begin{equation} \epsilon (1-u) + u\epsilon_s <0 \end{equation} \noindent It is convectively unstable for : \begin{equation} \epsilon (1-u) + u\epsilon_s >0 \end{equation} and absolutely unstable for : \begin{equation} \epsilon (1-u) - (\epsilon_c - u\epsilon_s )>0 \end{equation} Hence, for $u < 1$, pure rolls are absolutely unstable for any $\epsilon >\epsilon_s $ when $\epsilon_c<\epsilon_s $, while for $\epsilon_c> \epsilon_s $, they are convectively unstable and absolutely stable for $\epsilon_s <\epsilon< {\epsilon_c - u\epsilon_s \over 1-u}$, and absolutely unstable for ${\epsilon_c - u\epsilon_s \over 1-u}<\epsilon$. \medskip On the other hand, for $u > 1$ and $\epsilon_c>\epsilon_s $, they are convectively unstable and absolutely stable for any $\epsilon_s <\epsilon < {u\epsilon_s\over u-1}$, while for $\epsilon_c<\epsilon_s $, they are absolutely unstable for $\epsilon_s <\epsilon< {u\epsilon_s - \epsilon_c \over u - 1}$, and convectively unstable for ${u\epsilon_s - \epsilon_c \over u - 1}<\epsilon< {u\epsilon_s \over u - 1}$. In both cases, they are convectively stable for $ {u\epsilon_s\over u-1} < \epsilon $. \subsection{Stability of mixed states} The mixed states may be of two types, which result from superpositions of rolls and traveling wave states or rolls and standing wave states. \paragraph{Mixed traveling waves and roll states} These states are asymptotic solutions of the equations: \begin{eqnarray}\label{CGL3} \partial_T A &=& V \partial_X A + \epsilon A + (1+i\alpha) \partial_X^2 A - (1+i \beta)\vert A\vert^2 A \nonumber \\&-& u(1+i v)\vert R\vert^2 A \nonumber \\ \tau \partial_T R&=& (\epsilon -\epsilon_s )R + \xi_0^2 \partial_X^2 R - \vert R\vert^2 R \nonumber \\&-& w(1+i\zeta)\vert A\vert^2 R \end{eqnarray} with B=0 (or the symmetric states where A and B are exchanged). Their uniform solutions may be written as $A=\vert A_m\vert e^{i\phi_{A_m}}$, $R=\vert R_m\vert e^{i\phi_{R_m}}$, and satisfy: \begin{eqnarray}\label{CGL4} \epsilon &-& \vert A_m\vert^2 - u\vert R_m\vert^2 = 0 \nonumber \\ \epsilon -\epsilon_s &-& \vert R_m\vert^2 - w\vert A_m\vert^2 = 0 \nonumber \\ \phi_{A_m} &=& -(\beta \vert A_m\vert^2 + uv \vert R_m\vert^2)t\nonumber \\ \phi_{R_m}&=& -w\zeta \vert A_m\vert^2 t \end{eqnarray} As a result, one has : \begin{eqnarray} \vert A_m\vert^2 &=& {\epsilon (1-u) + u\epsilon_s \over 1-uw} \quad , \quad \nonumber \\ \vert R_m\vert^2 &=& {\epsilon (1-w) - \epsilon_s \over 1-uw} \end{eqnarray} The positivity of the norms require that $uw < 1$, $\epsilon (1-u) + u\epsilon_s > 0$, and $\epsilon (1-w) - \epsilon_s > 0 $. Hence, these solutions never exist for $w > 1$, while for $w < 1$, they exist for all $\epsilon >{\epsilon_s \over 1-w}$, if $u < 1$, and for ${\epsilon_s \over 1-w}<\epsilon<{u\epsilon_s \over u-1}$, for $11$, this state is stable, while for $\gamma >1$, it is absolutely unstable, except for \begin{equation} {\epsilon_s \over 1-w}<\epsilon < { \epsilon_c'(1-uw) - u\epsilon_s \over 1 - u} \end{equation} when $u < 1$, and for \begin{equation} {u\epsilon_s - \epsilon_c'(1-uw) \over u - 1}<\epsilon < {u\epsilon_s \over u-1} \end{equation} when $1{1+\gamma \over 2}$, while for $w <{1+\gamma \over 2}$, they exist for ${(1+\gamma)\epsilon_s \over 1+\gamma - 2w}<\epsilon $ when $u<1$, and for $\epsilon_w ={(1+\gamma) \epsilon_s \over 1+\gamma - 2w}<\epsilon < {u\epsilon_s \over u-1} =\epsilon_u$ when $11$, these states are unstable, while for $\gamma <1$ they are amplitude stable, but could be phase unstable, according to the value of the imaginary parts of the kinetic coefficients. In the following, I will consider them as stable, which is the case when the imaginary parts of the kinetic coefficients are sufficiently small. \section{Pattern selection in deterministic systems}\label{detpatsel} In this discussion, I will consider separately the cases where $\gamma>1$ which favors traveling waves, and where $\gamma$ vary in the range $-1<\gamma <1$, which implies supercritical bifurcations and preferred standing wave solutions. When $\gamma>1$, in the absence of group velocity, TW may develop for any $\epsilon > 0$. For $w>1$, they remain stable versus spatial modulations, although roll patterns may also develop in the range $\epsilon > {u\epsilon_s \over u-1}$ , when $u>1$. For $w<1$, TW states lose stability versus spatial modulations at $\epsilon ={\epsilon_s\over 1-w}$ where they bifurcate to rolls, when $uw>1$, or mixed modes when $uw<1$. In the presence of group velocity, TW may only be sustained by the dynamics for $\epsilon >\epsilon_c$. For $w>1$, they remain stable versus spatial modulations for all $\epsilon >\epsilon_c$, while rolls are stable in the range $\epsilon_s<\epsilon < {\epsilon_c - u\epsilon_s\over 1-u}$ when $u<1$, and for $\epsilon_s<\epsilon$ (if $\epsilon_s<\epsilon_c$), or ${ u\epsilon_s - \epsilon_c \over 1-u}<\epsilon $ (if $\epsilon_s>\epsilon_c$) when $u>1$. For $w<1$, TW states again lose stability versus spatial modulations at $\epsilon ={\epsilon_s\over 1-w}$ where they bifurcate to rolls, when $uw>1$, or mixed modes when $uw<1$. They are thus stable in the range $\epsilon_c<\epsilon<{\epsilon_s\over 1-w}$ . Let me consider now weak cross-couplings such that $-1<\gamma<1$. When $w> {1+\gamma \over 2}$ , standing waves may develop, in this case, for any $\epsilon >0$ in the absence of group velocity (bistability with steady rolls may occur for $\epsilon >\epsilon_s {u\over u -1}$ if $u>1$). For $w< {1+\gamma \over 2}$ , however, standing waves are stable up to $\epsilon =\epsilon_s {1+\gamma\over 1+\gamma -2w}$ where it bifurcates to mixed modes if $u<{1+\gamma\over 2w}$ or to rolls if $u>{1+\gamma\over 2w}$. In the range $1{1+\gamma\over 2}$, which prohibits the existence of mixed states. In such systems, one has to distinguish further between the $u<1$ and $u>1$ cases, with $\epsilon_c<\epsilon_s $ or $\epsilon_s <\epsilon_c$. When $w>{1+\gamma\over 2}$, and $\epsilon_c<\epsilon_s $, the trivial conduction state remains convectively unstable up to $\epsilon =\epsilon_c$ where it bifurcates to standing waves (although steady rolls could in principle appear for any $\epsilon >\epsilon_s $, they can only be sustained by the dynamics for $u>1$ in the range $\epsilon_s {u\over u -1} <\epsilon $, while they are absolutely unstable for $u<1$). In quench experiments, standing waves may appear as soon as $\epsilon'_c <\epsilon$ \medskip On the contrary, when $w>{1+\gamma\over 2}$ and $\epsilon_c>\epsilon_s $, the trivial conduction state remains convectively unstable up to $\epsilon =\epsilon_s $ where it bifurcates to steady rolls. For $u<1$, these rolls are convectively unstable up to $\epsilon = {\epsilon_c - u\epsilon_s \over 1-u}$ where they become absolutely unstable and bifurcate to standing waves. Since standing waves are stable as soon as $\epsilon >\epsilon_c$, the system is bistable in the range $\epsilon_c <\epsilon < {\epsilon_c - u\epsilon_s \over 1-u}$. For $u>1$ rolls are convectively unstable for $\epsilon_s<\epsilon <\epsilon_s {u\over u -1}$ and stable for $\epsilon_s {u\over u -1}<\epsilon $. The corresponding phase diagrams are represented in figures 3 and 4. \medskip\noindent For $w<{1+\gamma \over 2}$, the previous results are modified as follows. \medskip (1) For $\epsilon_c<\epsilon_s $ and $u<1$, the trivial conduction state is convectively unstable up to $\epsilon =\epsilon_c$ where it bifurcates to standing waves. These standing waves are stable up to $\epsilon =\epsilon_s {1+\gamma \over 1+\gamma -2w}$ where they bifurcate to a mixed standing waves/rolls state. \medskip (2) For $\epsilon_c<\epsilon_s $ and $1\epsilon_c$, the deterministic pattern selection should not be affected by the presence of noise since all the possible patterns are intrinsically sustained by the dynamics. On the other hand, standing or traveling waves should be sustained by spatially distributed noise for $0<\epsilon <\epsilon_c$ in ramp experiments starting from the trivial conducting state. For quench experiments in the range $\epsilon_s <\epsilon <\epsilon_c$, a competition may arise between noise sustained wave patterns and dynamically sustained roll patterns which needs to be studied numerically, as shown in section \ref{numerical} \medskip On noise removal, interesting situations may occur in systems where $\epsilon_s <\epsilon_c' <\epsilon_c$ since the standing waves should relax to traveling waves for $\epsilon_c'<\epsilon <\epsilon_c$, to steady rolls for $\epsilon_s <\epsilon <\epsilon_c'$, and to the conducting state for $0<\epsilon <\epsilon_s $. \section{Numerical analysis}\label{numerical} The preceding results have been confirmed by numerical tests performed with a finite difference code for a system of 200 points. The boundary conditions where $A(0) = R(0) = B_x(0) = 0$ and $A_x(200) = R(200) = B(200) = 0$ . For the stochastic cases, noise intensities have been chosen between $10^{-4}$ and $5.10^{-3}$. The following observations are particularly relevant : 1) In a system where $\gamma = -0.5 $ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w = 0.5$, $u = 2 $ and $V = 1$, the following succession of patterns is obtained in a ramp experiment (e.g. line R in fig.3) in a deterministic system : uniform steady state up to $\epsilon =\epsilon_s$ (e.g. point 1 on line R in fig.3) and rolls for $\epsilon_s<\epsilon$ (e.g. points 2 and 3 on line R in fig.3). In stochastic systems with spatially distributed noise , standing waves form for any $\epsilon >0$, but, on noise removal, the system relaxes to standing waves for$\epsilon'_c<\epsilon$ (e.g. point 3 on line R in fig.3) , and to rolls for $\epsilon_s<\epsilon < \epsilon'_c$ (e.g. point 2 on line R in fig.3). The corresponding numerical results are presented in figs. 8-10. 2) In systems where $\gamma = -0.5 $ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta = 0$, $w = 0.5$, $u = 0.75 $ and $V = 1$, ramp experiments (cf. line R in fig.4) lead to the following succession of patterns in deterministic system : uniform steady state up to $\epsilon =\epsilon_s$, rolls for $\epsilon_s<\epsilon < {\epsilon_c - u\epsilon_s \over 1-u}$ and standing waves for $\epsilon > {\epsilon_c - u\epsilon_s \over 1-u}$ while in stochastic systems with spatially distributed noise, standing waves form for any $\epsilon >0$. 3) In systems where $\gamma = 0.5 $ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w = 1.5$, $u = 2 $ and $V = 1$, ramp experiments (e.g. line R in fig.7) lead to the following succession of patterns in a deterministic system : uniform steady state up to $\epsilon =\epsilon_s$ and rolls for $\epsilon_s<\epsilon $ (e.g. points 1 and 2 on line R in fig.7), while in stochastic systems with spatially distributed noise, standing waves form for any $0<\epsilon $. However, after noise removal, the system relaxes to rolls for $\epsilon_s<\epsilon <\epsilon_c$ (e.g. point 1 on line R in fig.7) , and to traveling waves for $\epsilon_c<\epsilon <\epsilon'_c$ (e.g. point 2 on line R in fig.7). The corresponding results are presented in figs. 11 and 12. I have furthermore tested the effect of group velocity variations in deterministic systems where $\gamma = -0.5 $ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w = 0.2$, $u = 1.2 $. On increasing the group velocity from $V = 0.5$ to $V = 1.5$ (e.g. states 1 to 3 on line R in fig.5b), one passes from standing waves ($V = 0.5$ and $V = 1$) to rolls ($V = 1.5$), while on decreasing the group velocity from $V = 1.5$ to $V = 0.5$, the system remains in the rolls state confirming the bistability of rolls and waves patterns (cf. fig.13). On the other hand, interesting competition phenomena may occur between noise- and dynamically sustained structures. Although such competition is difficult to quantify and requires systematic numerical analysis, preliminary results show that the cross-coupling terms which renormalize the growth rate of the different kinds of modes could play a capital role in dynamical selection processes. For example, in a perfect co-dimension 2 situation ($\epsilon_s =0$), numerical results obtained for $\gamma = 0.5$, $u=2$ and small imaginary parts of the kinetic coefficients strongly differ for $wu/2$ in the presence of noise. For example, for $w=0.8$, roll structures emerge, irrespectively of the noise intensity, while for $w=1.5$, rolls or waves emerge at random for low noise intensities ($I<10^{-4}$), while waves always emerge at higher noise intensities ($I>10^{-4}$), as illustrated on figs. 14 and 15. The numerical results confirm that the healing length or boundary layer extension decreases for increasing noise intensity. On the other hand, for $w=1.5$, the stability of rolls obtained in a deterministic quench has been tested versus spatially distributed noise of increasing intensity. In such an experiment, once steady rolls are obtained, all the parameters of the dynamics are kept constant, except the noise intensity, which is slowly increased. Rolls turned out to be stable up to noise intensities of the order of $6.10^{-3}$, where they bifurcate to wave patterns (cf. fig. 16). These results confirm the subtle interplay between dynamic and stochastic effects on pattern selection in convectively unstable systems. Finally, it has to be noted that the amplitude of the patterns are also noise amplified. Effectively, although the moduli $\vert A\vert ^2$, $\vert B\vert ^2$ and $\vert R\vert ^2$ obtained numerically are found to agree with the analytical ones in deterministic systems, they are amplified in the presence of noise, as shown in fig. 17. In this case, rolls are sustained by the dynamics, and, in the absence of noise, the mean value of the modulus $<\vert R\vert ^2>$ reaches the expected deterministic value $\mu $ = 0.2, in the bulk. In the presence of spatially distributed noise, this value increases with noise intensity (for example, for a noise intensity of $8.10^{-3}$, $\vert R\vert ^2$ reaches 0.256), in agreement with the fact that the linear evolution of the mean square of the deviation of the roll amplitude around its deterministic value ($\rho = R -\sqrt{\mu}$) tends to an asymptotic value proportional to the noise intensity. \section{Conclusions}\label{conclusion} The conclusion of the analysis performed in this paper is that the presence of group velocity and mean flows may strongly affect pattern selection and stability in systems described by coupled Ginzburg-Landau equations, especially when the corresponding Hopf bifurcation is close to another instability which leads, for example, to steady roll patterns, as in the case of binary or viscoelastic fluid convection. When the absolute instability threshold of the trivial steady state remains below the stationary instability ($\epsilon_c <\epsilon_s$), the transition to wave patterns is only retarded by the mean flow in deterministic systems, contrary to stochastic ones where noise is able to sustain such patterns in the convectively unstable regime. In this case, pattern selection is thus not qualitatively modified by the neighboring stationary instability. On the contrary, in deterministic systems, when $\epsilon_c>\epsilon_s $, standing waves are eliminated as intermediate pattern between the conduction state and rolls or mixed modes, although bistability domains may exist. However, the succession of patterns that would occur in the absence of mean flow is recovered in the presence of spatially distributed noise, although interesting competition phenomena may occur between noise- and dynamically sustained structures. Preliminary numerical analysis show that, in this case, pattern selection should be very sensitive to the interplay between kinetic and stochastic effects on one side, and experimental protocols, on the other side. \bigskip\section*{Acknowledgments.} Financial assistance through a grant for sabbatical stay from the Ministerio de Education y Ciencia (DGICYT, Madrid) , and through a travel grant from the Belgian National Fund for Scientific Research is gratefully acknowledged. \begin{references} \bibitem[+]{Daniel} Director of Research at the Belgian National Fund for Scientific Research. 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San Miguel, Phys.Rev.Lett. {\bf 75}, 425 (1995). \end{references} \hrule \newpage . \vspace{1cm} \begin{figure}\label{fig1} \vspace{6cm} \special{psfile=fig1.eps hscale=55 vscale=55 hoffset=10} \bigskip \caption{Schematic phase diagram associated with the dynamical system (\ref{CGL2}), in the ($\epsilon$, $u$) plane, for $V =0$, $\gamma <0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w>{1+\gamma\over 2}$} \end{figure} \vfill \begin{figure}\label{fig2} \vspace{6cm} \special{psfile=fig2.eps hscale=55 vscale=55 hoffset=10} \bigskip \caption{Phase diagram associated with the dynamical system (\ref{CGL2}) for $V =0$, $\gamma <0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w<{1+\gamma\over 2}$} \end{figure} \bigskip . \vspace{1cm} \begin{figure}\label{fig3} \vspace{6cm} \special{psfile=fig3.eps hscale=50 vscale=50 } \bigskip \caption{Schematic phase diagram associated with the dynamical system (\ref{CGL2}) with non vanishing group velocity $V \ne 0$, in the ($\epsilon$, $\epsilon_c$) plane, for $\gamma <0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w>{1+\gamma\over 2}$ and $u>1$} \end{figure} \vfill \begin{figure}\label{fig4} \vspace{6cm} \special{psfile=fig4.eps hscale=50 vscale=50 } \bigskip \caption{Schematic phase diagram associated with the dynamical system (\ref{CGL2}) with non vanishing group velocity $V \ne 0$, in the ($\epsilon$, $\epsilon_c$) plane, for $\gamma <0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w>{1+\gamma\over 2}$ and $u<1$} \end{figure} \newpage . \vspace{1cm} \begin{figure}\label{fig5} \vspace{6cm} \special{psfile=fig5a.eps hscale=50 vscale=50 } \vfill \vspace{8cm} \special{psfile=fig5b.eps hscale=50 vscale=50 } \bigskip \caption{Schematic phase diagram associated with the dynamical system (\ref{CGL2}) with non vanishing group velocity $V \ne 0$, in the ($\epsilon$, $\epsilon_c$) plane, for $\gamma <0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w<{1+\gamma\over 2}$ and (a) ${1+\gamma\over 2w}0$ , $\alpha = 0.1$, $\beta = \delta = 0.15$ and $v = \zeta =0$ when $w>{1+\gamma\over 2}$ and $u>1$} \end{figure} \newpage . \bigskip \begin{figure}\label{fig8} \vspace{7cm} \special{psfile=TH1.eps hscale=45 vscale=45 voffset=-20} \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, for $\gamma = -0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=0.5$, $u= 2$, $\epsilon = 0.1$ and $\mu = \epsilon - \epsilon_s = -0.05$ (the transients disappear in the long time limit as the result of the convective instability of the trivial steady state)} \end{figure} \vfill \begin{figure}\label{fig9} \vspace{7cm} \special{psfile=TH2.eps hscale=45 vscale=45 } \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, for $\gamma = -0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=0.5$, $u= 2$, $\epsilon = 0.15$ and $\mu = \epsilon - \epsilon_s = 0.1$ } \end{figure} \bigskip . \bigskip \begin{figure}\label{fig10} \vspace{7cm} \special{psfile=TH3.eps hscale=45 vscale=45 voffset=-20} \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, for $\gamma = -0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=0.5$, $u= 2$, $\epsilon = 0.2$ and $\mu = \epsilon - \epsilon_s = 0.1$ } \end{figure} \vfill \begin{figure}\label{fig11} \vspace{7cm} \special{psfile=TH4.eps hscale=45 vscale=45 } \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, for $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=0.5$, $u= 2$, $\epsilon = 0.2$ and $\mu = \epsilon - \epsilon_s = 0.1$ } \end{figure} \newpage \bigskip . \bigskip \begin{figure}\label{fig12} \vspace{7cm} \special{psfile=TH5.eps hscale=45 vscale=45 voffset=-20} \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, for $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=1.5$, $u= 2$, $\epsilon = 0.3$ and $\mu = \epsilon - \epsilon_s = 0.2$ } \end{figure} \vfill \begin{figure}\label{fig13} \vspace{7cm} \special{psfile=TH6.eps hscale=45 vscale=45 } \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with varying group velocities, for $\gamma = -0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.15$, $v = \zeta =0$, $w=0.2$, $u= 1.2$, $\epsilon = 0.2$ and $\mu = \epsilon - \epsilon_s = 0.1$ } \end{figure} \bigskip . \bigskip \begin{figure}\label{fig14} \vspace{9cm} \special{psfile=weaknoise.eps hscale=45 vscale=45 voffset=-30} \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, in the presence of spatially distributed noise of low intensity ($I=2.10^{-4}$) , for $\epsilon = 0.2$, $\mu = \epsilon - \epsilon_s = 0.2$, $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.03$, $v = \zeta =0.07$, $u= 2$, and two different values of w ($w=0.8$ and $w=1.5$)} \end{figure} \vfill \begin{figure}\label{fig15} \vspace{7cm} \special{psfile=strongnoise.eps hscale=45 vscale=45 } \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, in the presence of spatially distributed noise of higher intensity ($I= 8.10^{-3}$) , for $\epsilon = 0.2$, $\mu = \epsilon - \epsilon_s = 0.2$, $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.03$, $v = \zeta =0.07$, $u= 2$, and two different values of w ($w=0.8$ and $w=1.5$)} \end{figure} \newpage . \bigskip \begin{figure}\label{fig16} \vspace{7cm} \special{psfile=noiseramp.eps hscale=45 vscale=45 voffset=-20} \bigskip \caption{Results of the numerical resolution of the dynamical system (\ref{CGL2}) with non vanishing group velocity $V = 1$, in the presence of spatially distributed noise of increasing intensity (I) ( $\epsilon = 0.2$, $\mu = \epsilon - \epsilon_s = 0.2$, $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.03$, $v = \zeta =0.07$, $u= 2$, $w=1.5$), showing a noise induced transition from roll to wave patterns. } \end{figure} \vfill \begin{figure}\label{fig17} \vspace{7cm} \special{psfile=noisyrolls.eps hscale=45 vscale=45 } \bigskip \caption{Modulus of the roll amplitude $\vert R\vert ^2$ obtained numerically from the dynamical system (\ref{CGL2}) in the presence of spatially distributed noise of increasing intensity (I) ( $\epsilon = 0.2$, $\mu = \epsilon - \epsilon_s = 0.2$, $\gamma = 0.5$ , $\alpha = 0.1$, $\beta = \delta = 0.03$, $v = \zeta =0.07$, $u= 2$, $w=0.8$, $V = 1$). } \end{figure} \end{document}