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the rayleigh - bnard convection problem is one of the fundamental problems in the physics of fluids .
the basic phenomena of the rayleigh - bnard convection in horizontally extended systems are widely known .
the influence of the side walls , although not studied as throughly as the horizontally extended case , is of practical importance for engineering applications .    in this paper
we study the rayleigh - bnard convection in a 2-d rectangular domain with no - slip boundary conditions for the velocity .
this problem is also closely related to the problem of infinite channel with rectangular cross - section which has been studied by davies - jones @xcite , luijkx 
platten @xcite and kato 
fujimura @xcite among others .
the linear aspects of the problem we consider in this paper have been studied by lee 
boyd @xcite , mizushima @xcite and gelfgat @xcite . in these papers , the critical rayleigh number and the structure of the critical eigenmodes
have been studied for small aspect ratio containers .    from dynamical transition and pattern formation point of view , ma and wang @xcite proved that under some general boundary conditions , the problem always undergoes a dynamic transition to an attractor @xmath4 as the rayleigh number @xmath5 crosses the first critical rayleigh number @xmath0 .
they also proved that the bifurcated attractor , homological to @xmath6 , where @xmath2 is the number of critical eigenmodes .    in the 2-d setting that we consider ,
@xmath2 is either 1 or 2 and the latter case can only happen at the critical length scales where two modes with wave numbers @xmath7 and @xmath8 become critical simultaneously .
when @xmath9 , the structure of @xmath4 is trivial which is merely a disjoint union of two attracting steady states .
thus our task in this paper is to classify the structure of the attractor when @xmath3 .
this has been studied recently in @xcite for the 3d rayleigh - bnard problem where the boundaries were assumed to be free - slip for the velocity and the wave numbers of the critical modes were assumed to be equal .
the main mathematical challenge in this paper is due to the no - slip boundary conditions since the separation of variables for the linear eigenvalue problem which works in the free - slip case is not possible anymore . to overcome this difficulty ,
the main approach for our study is to combine rigorous analysis and numerical computation using spectral method .
as we know , spectral methods have long been used to address the hydrodynamic instability problems .
in fact , in his seminal work @xcite , orszag studied the classical orr - sommerfeld linear instability problem using a chebyshev - tau method . in this paper , to treat the linear eigenvalue problem , we employ a legendre - galerkin method where compact combinations of legendre polynomials , called generalized jacobi polynomials , satisfying all the boundary conditions are used as trial functions .
the main advantage of our legendre - galerkin method is that the resulting matrices are sparse which allows a very efficient and accurate solution of the linearized problem ; see also hill 
straughan @xcite and gheorghiu  dragomirescu @xcite .
once the eigenpairs of the linear problem are identified , the transition analysis is carried out by reducing the infinite dimensional system to the center manifold in the first two critical eigendirections .
the coefficients of this reduced system are calculated numerically .
our main results are described below .
we first classify the eigenmodes into four classes according to their parities using the symmetry of the problem .
then we numerically show that the first two unstable modes are always parity class one or two
. then we study the transition near the critical length scales where two eigenvalues become positive simultaneously .
next , we rigorously prove that the local attractor at small supercritical rayleigh numbers is in fact homeomorphic to the circle which has four or eight steady states with half of them as stable points and the rest as saddle points
. the critical eigenmodes are always bifurcated steady states on the attractor and when the attractor has eight steady states , the mixed modes which are superpositions of the critical eigenmodes are also bifurcated .
second , let @xmath10 and @xmath11 denote the two largest eigenvalues of the linearized problem . we find that a small neighborhood of @xmath10 = @xmath12 in the @xmath10@xmath11 plane can be separated into several sectors with different asymptotical structures . in particular , we find that there is a critical prandtl number @xmath13 for the first two critical length scales @xmath14 and @xmath15 , such that for @xmath16 , there is a sector in this plane for which mixed modes are stable fixed points of the attractor . for @xmath17 ,
the mixed modes are never stable and instead there is a sector in this plane in which both of the critical eigenmodes coexist as stable steady states . in this case , the initial conditions determine which one of these eigenmodes will be realized .
the critical prandtl number is around @xmath18 for the first critical length scale @xmath14 and around @xmath19 for the second critical length scale @xmath15 . for higher critical length scales we found that mixed modes are never stable points of the attractor .
third , recently ma 
wang has developed the dynamic transition theory to study transition and bifurcation problems in nonlinear sciences ; see @xcite .
this paper is a first attempt to combine this theory with the numerical tools of the spectral methods to study the detailed structure of the transition and pattern formation .
the paper is organized as follows . in section 2 , the governing equations and the functional setting of the problem is discussed . in section 3 ,
linear eigenvalue problem is studied .
section 4 states the main theorem .
section 5 is devoted to the proof of the main theorem . in section 6 ,
we demonstrate a method to compute the coefficients of the reduced system .
and the last section discusses the results obtained by our analysis .
two dimensional thermal convection with no - slip , perfectly conducting boundaries can be modeled by the boussinesq equations . the governing equations on the rectangular domain @xmath20 read as @xmath21 here @xmath22 is the velocity field , @xmath23 is the temperature field and @xmath24 is the pressure field .
these fields represent a perturbation around the motionless state with a linear temperature profile .
the dimensionless numbers are the prandtl number pr and the rayleigh number r which is also the control parameter .
@xmath25 represents the unit vector in the z - direction .
the equations are supplemented with no - slip boundary conditions for the velocity and perfectly conducting boundary conditions for the temperature .
@xmath26    for the functional setting ,
we define the relevant function spaces : @xmath27 for @xmath28 , let @xmath29 and @xmath30 be defined by @xmath31 with @xmath32 denoting the leray projection onto the divergence - free vectors .
the equations  supplemented with initial conditions can be put into the following abstract ordinary differential equation : @xmath33    for results concerning the existence and uniqueness of solutions of , we refer to foias , manley , and temam @xcite .    finally for @xmath34 , @xmath35 , @xmath36
we define the following trilinear forms . @xmath37
we first study the eigenvalue problem @xmath38 which reads as @xmath39 below we list some of the properties of this eigenvalue problem .    *
the linear operator @xmath40 is symmetric .
hence the eigenvalues @xmath41 are real and the eigenfunctions @xmath42 are orthogonal with respect to @xmath43inner product .
moreover there is a sequence @xmath44 such that @xmath45 .
@xmath46 is found by setting @xmath47 in . in this case
the problem becomes an eigenvalue problem with @xmath48 as the eigenvalue .
* we have @xmath49 which can be seen by computing the derivative of @xmath41 with respect to @xmath5 at @xmath50 .
@xmath51 where @xmath52 is the ith eigenfunction . also at @xmath50 , by the third equation in @xmath53 as @xmath54 . plugging these into and integrating by parts
, we see that @xmath55 * we denote the critical rayleigh number @xmath56 .
that is @xmath57 @xmath2 in does not depend on the prandtl number pr but only on @xmath58 . to see this ,
one makes the change of variable @xmath59 so that the solution of for the eigenvalue of @xmath47 is independent of pr . by simplicity of the first eigenvalue ( see theorem 3.7 in ma  wang @xcite ) , for almost every value of @xmath58 except a discrete set of values , @xmath2 in is 1 .
the linear equations satisfy several discrete symmetries which may be found from the known non - trivial groups of continuous lie symmetries of the field equations ; see ( hydon@xcite , marques  lopez 
blackburn @xcite )
. however , for the problem we consider , it can be easily verified that the linear equations have reflection symmetries about the horizontal and vertical mid - planes of the domain .
thus we can classify the solutions of the linear problem into four classes with different parities which are as defined in table  [ tab : parity ] where , for example , @xmath63 means that @xmath64 is odd in the @xmath65-direction and even in the @xmath66-direction .                      y.  kato and k.  fujimura , _ prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow through a rectangular channel heated from below _
, physical review e * 62 * ( 2000 ) , no .  1 , 601 .
we study the rayleigh - bnard convection in a 2-d rectangular domain with no - slip boundary conditions for the velocity . 
 the main mathematical challenge is due to the no - slip boundary conditions , since the separation of variables for the linear eigenvalue problem which works in the free - slip case is no longer possible . 
 it is well known that as the rayleigh number crosses a critical threshold @xmath0 , the system bifurcates to an attractor , which is an @xmath1dimensional sphere , where @xmath2 is the number of eigenvalues which cross zero as r crosses @xmath0 . 
 the main objective of this article is to derive a full classification of the structure of this bifurcated attractor when @xmath3 . 
 more precisely , we rigorously prove that when @xmath3 , the bifurcated attractor is homeomorphic to a one - dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits . 
 in addition , we show that the mixed modes can be stable steady states for small prandtl numbers .