Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS/Université de Lyon, Ecole Centrale de Lyon/Université Lyon 1/INSA de Lyon,ECL, Ecully,France
Corresponding author: D. Henry, Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS, Université de Lyon, Ecole Centrale de Lyon, Université Lyon 1, INSA de Lyon, ECL, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France. (firstname.lastname@example.org)
Corresponding author: D. Henry, Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS, Université de Lyon, Ecole Centrale de Lyon, Université Lyon 1, INSA de Lyon, ECL, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France. (email@example.com)
 The convection induced by buoyancy effects in a porous square cavity has been investigated numerically using a spectral element code with bifurcation tools. The complex bifurcation diagram initiated from the first six primary bifurcation points corresponding to the onset of convection from the trivial no-flow solution has been calculated. Four branches of stable steady solutions have been found, corresponding to one-roll, two-roll, three-roll and four-roll flow structures. The domain of existence of these stable solutions, i.e., the Rayleigh number (Ra) range in which such solutions can potentially be observed, has been precisely determined. It is shown that there exist Ra ranges where different flow solutions can be stable together. The stable branches all terminate at Hopf bifurcation points beyond which oscillatory solutions have been computed.
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 In many porous natural and industrial systems, variable density effects play an important role, as they drive the flow patterns and thus strongly influence the associated mass or heat transfer processes. Several scientific areas are concerned by these phenomena such as hydrology, geophysics, oil reservoirs study, nuclear waste storage, CO2 sequestration, etc. [Diersch and Kolditz, 2002]. The problems studied in this context are typically related to the flow of brines with variable concentration in deep or near-surface rock formations. Due to its practical importance for groundwater resources management, the problem of seawater intrusion into coastal aquifers (first modeled by Henry ) can be considered as the first problem that interested the hydrogeologic community in this field. Despite the relatively low concentration gradients in this configuration, the questions raised about the mathematical formulation of the physical processes, the geometry of the domain, the parameterization, the boundary conditions and the numerical methods to be applied in this case, have been widely discussed since then and until the 1980s [Pinder and Cooper, 1970; Caltagirone et al., 1987; Herbert et al., 1988; Hassanizadeh and Leijnse, 1988; Diersch, 1988]. New steps in this research field have been connected with the development of new technologies, as the nuclear waste storage in deep geological formations of low permeability and the CO2 sequestration in deep saline aquifers. The new scientific questions thus raised concern the possibility of high density and viscosity contrasts, the presence of reactive multispecies, dissolution-precipitation phenomena which affect the solid matrix [Bouhlila, 1999], or the presence of two phases [Liu et al., 2011].
 In a context of nonreactive mass transfer, the density-dependent flows in porous media are induced by the spatiotemporal variations of the fluid density depending on a scalar field (the temperature or the solute concentration), which affect the buoyancy term in the Darcy equation or other more sophisticated momentum conservation equations. To solve the nonlinear equations that finally govern the coupled processes of flow and scalar field transport in porous media, sophisticated numerical methods are required. Besides, multiple solutions can exist and only one is obtained during a given time-evolution simulation [Diersch and Kolditz, 2002]. A typical case mentioned in the review of Diersch and Kolditz  is the Horton-Rogers-Lapwood (HRL) problem which is the analog of the Rayleigh-Bénard convection for porous media. Although it is a simplified situation, the related results, often expressed in terms of critical Rayleigh numbers , can shed light on other more complex and more real configurations. Weatherill et al.  also propose to use this problem as a test case for density-dependent groundwater flow and solute transport simulators.
 A linear stability analysis of the pure conduction base state in the HRL problem was carried out by Sutton . The critical thresholds for a square cavity, given by , correspond to perturbation eigenmodes with m horizontal cells and n vertical cells. The well known first critical Rayleigh number is given by and it gives rise to the development of a one-cell convective regime ( ). This one-cell regime has been shown to be stable up to where a fluctuating regime appears [Caltagirone, 1975]. Although the multicellular flows ( horizontal cells, n = 1) are unstable at onset from the base state, they have been obtained numerically for different values of , even in the range ( ) where a one-cell oscillatory flow exists [Horne and O'Sullivan, 1974]. The flow structure obtained depends on the initial conditions, the rate of heating, the perturbations applied. Partial heating of the lower boundary forces the solution as a unique mode of flow [Horne and O'Sullivan, 1974], whereas a temperature perturbation locally applied at this lower boundary can drive the oscillatory solution to either symmetric or asymmetric one-cell convection [Horne and Caltagirone, 1980]. Straus, Schubert and coworkers [Straus and Schubert, 1979; Schubert and Straus, 1979, 1982; Kimura et al., 1986, 1989] considered both the square and cubic configurations. For a cubic box, they discussed the possible occurrence of either two-dimensional or three-dimensional flows depending on the value of the Rayleigh number [Straus and Schubert, 1979; Schubert and Straus, 1979]. For a square cavity, they studied the time-dependent single-cell convection [Schubert and Straus, 1982; Kimura et al., 1986]. According to their results, with the increase of the Rayleigh number convection evolves from the initially periodic flow to a quasiperiodic flow, then followed by a second periodic flow, before the occurrence of a chaotic flow. Similar sequences of bifurcations are found in the case of a cubic cavity [Kimura et al., 1989].
 In our paper, we want to revisit the Horton-Rogers-Lapwood problem in a porous square box using nonlinear numerical analyses appropriate for the study of multiple solution problems. A nice paper on this problem, using similar analyses, has been published by Riley and Winters . They studied the complex bifurcation structure of the problem and, using the dynamical system theory, they elucidated the interactions between the different branches of solutions for cavities with an aspect ratio between 1 and 2. Our objective is to extend this study (limited to Rayleigh numbers roughly below 400) to a much larger range of Rayleigh numbers in order to capture all the different stable steady solutions and to precisely determine their existence range. A continuation method allowing to follow the solution branches, to calculate their stability and to precisely locate the different bifurcation points is used for that, and the results obtained are shown as bifurcation diagrams. This method is described in section 3, just after the presentation of the model in section 2. The results are then given in section 4. Four stable steady solutions have been found corresponding to one to four rolls along the horizontal. These solutions are eventually destabilized at Hopf bifurcation points beyond which oscillatory solutions have been computed.
 We want to determine the convection induced in a liquid confined in a porous square box heated from below. This box has a height h and a length l = h. The temperature is fixed to at the bottom boundary of the box and to at the top boundary, with , and there is no heat flux through the vertical boundaries. The normal derivative of the velocity is zero on all these boundaries. In such a situation, there exists a basic diffusive solution corresponding to no flow and to a linear temperature profile varying from to along the height of the box. As the density generally decreases with the increase of the temperature , such a situation with a stronger density at the top than at the bottom is potentially unstable and can lead to the onset of convective motions if the temperature difference is strong enough (beyond a threshold value). Finally, the flow in the porous medium will be modeled through the simple Darcy model. To solve this buoyancy induced convection problem, we then have to consider the following system of equations corresponding to the Darcy equation, the mass conservation equation and the heat conservation equation:
where k is the permeability of the porous medium, is the dynamical viscosity of the fluid, is its density, and is its thermal diffusivity. , and are the dimensional velocities, pressure and temperature, respectively. The fluid is assumed to be Newtonian with constant physical properties, except for the density in the buoyancy term which, in the Boussinesq approximation, depends linearly on the temperature,
where is the thermal expansion coefficient, is the mean temperature, , and is the value of the density at . We will use h, , , , and , as scales for length, velocity, time, pressure and temperature, respectively, and define the dimensionless temperature as ( varies from 0.5 to −0.5 when varies from to ). We then obtain the following system of dimensionless equations:
where the nondimensional parameter is the Rayleigh number defined as
is the dimensionless velocity and p is the dimensionless pressure. is the unit vector in the vertical z direction, and the horizontal direction corresponds to the x direction. The boundary conditions are given by
If the geometry of the two-dimensional cavity is fixed (square cavity), the only free parameter is the Rayleigh number . As a first step, for selected values of , we will compute some steady state solutions by time-evolution calculations. To have a better view of the solutions, we will then use a continuation method which will allow to obtain the global bifurcation diagram of our convective system.
 The symmetries of our system and of the basic diffusive state are an up-down reflection symmetry SH with respect to the horizontal middle plane (H plane) and a left-right reflection symmetry SV with respect to the vertical middle plane (V plane). These symmetries are defined, respectively, as:
The combination of these two symmetries gives a symmetry SC with respect to the center point of the cavity ( ) defined by
When increasing , bifurcations to convective flow states (steady or oscillatory) will occur, at which some of these symmetries will usually be broken.
3. Numerical Methods
 The governing equations of the model were solved in the two-dimensional domain using a spectral element method, as described by Ben Hadid and Henry . The spatial discretization is obtained through Gauss-Lobatto-Legendre points distributions; the time discretization is carried out using a semi-implicit splitting scheme, as proposed by Karniadakis et al.  but adapted to take into account the more simple Darcy equation. For the temperature , the nonlinear terms are first integrated explicitly and the linear terms are then integrated implicitly. The pressure p is then solved through a pressure equation enforcing the incompressibility constraint (with a consistent pressure boundary condition). The velocity field is then deduced from (5) using and p. This time integration scheme is used for transient computations with the third-order accurate formulation described by Karniadakis et al. .
 One of our objectives is to compute the global bifurcation diagram of our system. For that, we are able to follow steady state solutions for increasing by using the Newton method described by Henry and Ben Hadid . Leading eigenvalues—those with largest real part and thus responsible for initiating instabilities—and their corresponding eigenvectors can then be determined using Arnoldi's method (ARPACK library [Lehoucq et al., 1998]) by time stepping the linearized equations, as described by Mamun and Tuckerman . When the real part of an eigenvalue changes sign as is increased, this is an indication that a bifurcation point just occurred. The corresponding solution and eigenvector are thus used as initial guesses in the direct calculation of the bifurcation point, which is performed using the Newton method described by Henry and Ben Hadid . In the Newton methods used for both steady state solving and threshold calculations, the main idea is to solve the linear systems appearing at each Newton step with an iterative solver, and to compute right hand sides and matrix-vector products corresponding to these linear systems by performing adapted first-order time steps of the basic or linearized problem. The advantage of this method is that the Jacobian matrix does not need to be constructed or stored. The GMRES algorithm from the NSPCG software library [Kincaid et al., 1989] is used as the iterative solver.
 In practice, a continuation procedure organizes the different steps of the calculation. The leading eigenvalues of the diffusive solution are first calculated by Arnoldi's method for increasing in order to detect the primary bifurcations, which are then precisely determined by a direct calculation. For each primary bifurcation point, successively, a branching algorithm uses the critical eigenvector in order to jump to the emerging solution branch. This branch is then followed by continuation up to a chosen limit number. Along the branch, the leading eigenvalues are regularly calculated by Arnoldi's method in order to detect and then precisely locate secondary (steady or Hopf) bifurcation points. The steady branches emerging at the steady secondary bifurcation points can in turn be followed, if it is necessary to understand the dynamics of the system.
 Different convergence tests have been performed in order to determine the grid size necessary for our calculations of convective flows in the square cavity. These tests which concern the critical Rayleigh number for different bifurcation points are shown in Table 1. We see that the position of the first primary bifurcation point is already obtained with a great precision with a mesh, but that a slightly more refined mesh with points is necessary to have a good precision on the fifth primary bifurcation point. Concerning the secondary bifurcation points, they are quite well estimated with a mesh. Finally, concerning the Hopf bifurcation points which generally occur for larger values of , we see that the first point on the first primary branch can be quite well estimated with a mesh, but that the first point on the third primary branch needs a very refined mesh with points to be well calculated. In practice, we have chosen to calculate the first bifurcation diagram which is limited to with a mesh, whereas the continuation of the stable branches beyond is performed with a mesh. In the case of the four-roll branch, we even had to choose a mesh to ensure a good precision. Note finally that for , all the characteristic values given in the text as well as the flow and eigenvector patterns plotted in the figures have been recalculated with a refined mesh.
Table 1. Precision Tests on the Calculation of the Critical Rayleigh Number for Different Bifurcation Pointsa
First primary bifurcation point (Prim1), fifth primary bifurcation point (Prim5), first secondary bifurcation point on the second primary branch (Sec1Br2), second secondary bifurcation point on the fifth primary branch (Sec2Br5), first Hopf bifurcation point on the first primary branch (Hopf1Br1), and first Hopf bifurcation point on the third primary branch (Hopf1Br3).
11 × 11
17 × 17
21 × 21
23 × 23
51 × 51
81 × 81
91 × 91
4.1. Steady Solutions Obtained by Time-Evolution
 As a first step, we compute steady solutions of our two-dimensional convective system by performing time-evolution calculations with the time-stepping code. These calculations are initiated with the diffusive temperature profile which is perturbed by different types of temperature profiles applied at midheight of the cavity where the diffusive temperature is zero. These temperature profiles are chosen so that they could initiate different multiroll structures. For one-, two-, three-, and four-roll structures, we chose profiles proportional to , , , and , respectively. The calculations are performed for selected values of ranging from to .
 For , no flow can be triggered, which means that this situation is below the critical threshold for the onset of convection, theoretically predicted as . For , with any perturbation, the time evolution leads to a flow corresponding to one convective cell, which confirms that this situation is above the critical threshold. Other simulations have been performed for , 200, 300, and 500. Depending on the applied perturbation, different types of solutions have been obtained for these values of , which are denoted as (1), (2), (3), and (4) and correspond to structures with one roll, two rolls, three rolls, and four rolls across the horizontal length, respectively. We have generally applied perturbations with an amplitude equal to 0.1 (multiplicative coefficient modulating the selected profiles). As we can see in Figures 1–234 where isolines of temperature for the different steady solutions obtained are plotted, for we can get the solutions (1) and (2), for and we can get the solutions (1), (2), and (3), and for we can get the solutions (2), (3), and (4). Except for the four-roll solution, all these solutions have been obtained for an amplitude equal to 0.1, but can also be obtained with a smaller amplitude as 0.01. Concerning the four-roll solution, it has been obtained for with a perturbation amplitude equal to 0.2, but was not obtained for the other values even when the amplitude was increased to 0.5. The three-roll solution was also not obtained for . Finally, for an oscillatory one-roll solution was also found. We also did some calculations with a random temperature profile applied at midheight. For , one-roll and two-roll structures have been triggered, whereas for and 300, mainly two-roll structures were obtained. From these results, we clearly see that, in a certain range of , several different steady solutions can be obtained at the same value.
 Our objective is to get more information on this multiplicity of solutions. We have first to note that for a given solution to be obtained at some Rayleigh number, this solution has first to be stable and then the initial conditions have to enable the flow to reach the basin of attraction of the solution. From the review of the literature and our time-evolution calculations, we see that it is very difficult to get information on the basin of attraction of a given solution. Only ponctual information can be obtained by a specific time-evolution calculation with specific initial conditions. Moreover, numerical methods allowing the calculation of basins of attraction in complex systems still do not exist. In contrast, the stability domain of a given solution, i.e., the range in which this solution can be potentially obtained, is the kind of information we can expect to get from the calculation of the bifurcation diagram by a continuation method. We will see that in sections 4.2 and 4.3.
4.2. Bifurcation Diagram for Ra ≤ 300
 Following the procedure described in section 3, we were able to obtain the bifurcation diagram of our two-dimensional convective system. We first present the main different steady solution branches which can be obtained for in Figure 5. In this bifurcation diagram, where the maximum absolute value of the horizontal velocity is plotted as a function of , the stable solutions are given as solid lines whereas the unstable solutions are given as dashed lines. Precisions on the stability of each branch are given using the notation n – p, where n is the number of unstable real eigenvalues (those which are positive) and p is the number of unstable complex conjugate eigenvalues (those with positive real parts). Thus the number of unstable eigenvalues is , and the corresponding branch is said to be ( )-time unstable. A stable branch has no unstable eigenvalues and corresponds then to . For simplicity, when p = 0 only n is given. Solid circles indicate steady bifurcation points and open circles indicate oscillatory (Hopf) bifurcation points. It should be noted that the position of these bifurcation points in terms of critical Rayleigh number is well defined and does not depend on the variable plotted in the bifurcation diagram. These bifurcation points really define the structure of the bifurcation diagram. Only the first six primary branches and the secondary branches which participate to the stabilization of these primary branches are shown in the figure, having in mind that our focus is principally to track the existence range of the stable solutions. A more comprehensive study of the interactions between the different modes in the range using the dynamical system theory can be found in the paper of Riley and Winters .
 The first six primary bifurcation points are plotted with solid circles on the horizontal axis of the bifurcation diagram in Figure 5. They are located at , , , , , and , these values being rounded with two digits past the decimal point, as will be generally done for the other critical Rayleigh numbers given in the paper. (The next primary bifurcation points, which are much farther, beyond , are not given, but they do not influence the dynamics of the first four stable solutions in our system.) We can check that the first point (more precisely computed at ) is very close to the theoretical value . The other points are also very close to the corresponding theoretical values, as can be seen in Table 2.
Table 2. Comparison Between the Computed and Theoretical Values for the First Six Primary Critical Rayleigh Numbersa
The given values have been rounded with four digits past the decimal point. The type of flow structure triggered at these critical points is indicated by the notation , where m (n) denotes the number of rolls along the horizontal (vertical). The theoretical thresholds are given by [Sutton, 1970; Riley and Winters, 1989].
 The eigenvectors associated with these first six primary bifurcation points are shown in Figure 6. Both isolines of temperature and velocity vectors are plotted for each eigenvector. As the basic state is a no-flow state, these eigenvectors will determine the flow structures which will appear on the primary branches, the solutions being further modified by nonlinear effects when is increased. We see that the first three eigenvectors correspond to one-roll, two-roll, and three-roll structures, respectively, structures which were already denoted as (1), (2), and (3) structures. The fourth eigenvector has two rolls in the horizontal direction, but also two rolls in the vertical direction. We can denote this structure as a (2,2) structure, the second number indicating the rolls in the vertical and being omitted when it is 1. The fifth eigenvector correspond to a four-roll structure already denoted as a (4) structure. Finally, the sixth eigenvector has three rolls in the horizontal direction and two in the vertical direction and will be denoted as a (3,2) structure. The (1), (2), (3), and (4) structures break the up-down SH symmetry, which is not the case for the (2,2) and (3,2) structures. In contrast, the left-right SV symmetry is broken by the (1), (3), and (3,2) structures. As a result, the (1) and (3) structures will keep the central symmetry SC, the (2) and (4) structures the left-right SV symmetry, and the (3,2) structure the up-down SH symmetry, whereas the (2,2) structure keeps all the symmetries. All the primary bifurcations with symmetry breakings will be pitchfork bifurcations, and two equivalent solutions, with opposite rotation of the rolls (one solution being in fact the symmetric of the other with respect to the broken symmetries) will emerge at these bifurcations. Due to the choice of as the ordinate of our bifurcation diagram, these two equivalent solutions will appear as a single curve in Figure 5. In contrast, the bifurcation to the (2,2) structure ought to be transcritical and give rise to a supercritical branch and a subcritical branch. As often, the hysteresis associated with this transcritical bifurcation is too small to be seen in the bifurcation diagram, but two different branches will emerge at this point.
 We now analyze the different branches of the bifurcation diagram (Figure 5).
 1. The one-roll primary branch which emerges supercritically at is stable and will remain stable in the range of shown in Figure 5 ( ).
 2. Beyond , the basic diffusive solution is one-time unstable, so that the two-roll primary branch which emerges supercritically at is also one-time unstable at onset. This branch is however stabilized at a secondary bifurcation point at (this point is given by Riley and Winters  as ), where a one-time unstable supercritical branch emerges. The new branch will remain one-time unstable for whereas the two-roll primary branch will remain stable.
 3. Similarly, the three-roll primary branch which emerges supercritically at is two-time unstable at onset, but will be stabilized thanks to two secondary bifurcations, the first one at and the second one at . (These secondary bifurcations are shown in the paper of Riley and Winters , but the corresponding values of are not given.) The first secondary bifurcation is transcritical and two different branches emerge at this point. One of these branches (shown in Figure 5) emerges subcritically but quickly undergoes a saddle-node bifurcation (at ) and reverses direction. One-time unstable before the saddle-node point, it becomes two-time unstable beyond this point in the range of studied here. In contrast, as shown by Riley and Winters , the supercritical branch connects with a bifurcation point on the (2,2) branch. This connection, shown in Figure 7, slightly differs from what is shown in the paper of Riley and Winters . The supercritical branch, two-time unstable at onset, becomes one-time unstable beyond a steady bifurcation point at , but is further destabilized at a Hopf bifurcation point at and at a saddle-node bifurcation point at . Beyond this saddle-node point where the branch reverses direction, two steady eigenvalues and a pair of complex conjugate eigenvalues are unstable. The complex conjugate eigenvalues quickly collide (for ) on the positive real axis, so that the branch becomes four-time unstable. This branch will finally connect with the (2,2) branch at after a last steady bifurcation which occurs at . Concerning the second secondary branch on the three-roll primary branch (Figure 5), it emerges supercritically and will stay one-time unstable.
 4. The fourth primary branch corresponding to a (2,2) structure will in fact correspond to two different branches. The branch given in the diagram in Figure 5 is three-time unstable at onset at and will become four-time unstable beyond a secondary bifurcation point at . The other branch (with opposite rotation of the rolls) is given in Figure 7. Although its path is very close to that obtained for the first branch, this second branch is different, as indicated by the different stability properties. Three-time unstable close to onset, it becomes two-time unstable at a secondary bifurcation point at (where the connection with the three-roll branch occurs), but the stabilized eigenvalue collides soon for with another stable eigenvalue to give a stable complex conjugate eigenvalue. Another steady eigenvalue is destabilized at , before the destabilization of the complex conjugate eigenvalue at . Beyond this point, the branch then has three steady eigenvalues and one complex conjugate eigenvalue which are unstable.
 5. The changes which affect the four-roll primary branch which emerges supercritically at are still a little more complex. (This branch is not mentioned in the paper of Riley and Winters .) This branch, four-time unstable at onset, becomes five-time unstable at a secondary bifurcation point at . Two of these unstable steady eigenvalues will collide for to give a complex conjugate eigenvalue, and two others will do the same for . These two complex conjugate eigenvalues will be stabilized successively at Hopf bifurcation points at and . Finally, the last unstable eigenvalue is stabilized at a secondary bifurcation point at , and the four-roll primary branch thus becomes stable beyond this point. The first secondary branch initiated at is five-time unstable at onset but quickly becomes four-time unstable. After different collisions between the eigenvalues, the branch finally has two steady eigenvalues and one complex conjugate eigenvalue which are unstable at . The second secondary branch, which is one-time unstable at onset, has a complex path with bifurcation and saddle-node points, but is again simply one-time unstable at .
 6. The sixth primary branch corresponding to a (3,2) structure emerges at at a bifurcation which seems to be very slightly subcritical. Close to onset, this branch is five-time unstable. It becomes four-time unstable at a secondary bifurcation point at , but again five-time unstable at another secondary bifurcation point at . Two of the eigenvalues then collide to give a complex conjugate eigenvalue, so that at there are three steady eigenvalues and one complex conjugate eigenvalue which are unstable.
 From the analysis of the bifurcation diagram, we found that four steady solutions belonging to four of the primary branches are stable at . These stable solutions are shown in Figure 8. They correspond to one-, two-, three-, and four-roll flow structures which were initiated at the first, second, third and fifth primary bifurcation points, respectively. These stable solutions are completely similar to those obtained previously by time-evolution calculations (in some cases, a solution could be replaced by its symmetric, but, as already said, both solutions, initiated at the same pitchfork bifurcation, are completely equivalent). Note, however, that by time-evolution calculations we did not succeed to get the four-roll structure, even by perturbating the basic diffusive state with ad-hoc temperature perturbations.
 The two solutions initiated at the fourth and sixth primary bifurcations, which remain unstable, are plotted for in Figure 9. We see the completely symmetric structure of the solution on the fourth primary branch (Figure 9a) with four symmetric counter-rotating rolls (two along x and two along y) and the only up-down symmetric structure of the solution on the sixth primary branch (Figure 9b) with six rolls, three along x and two along y. Indeed, these flow structures are unstable and cannot be obtained by time-evolution calculations.
4.3. Existence Range of the Stable Steady Solutions
 We have extended the bifurcation diagram given in Figure 5 by following the main solution branches up to much larger values of . In that way, we were able to find the limit of stability of the branches which were stable at and to check that the other branches remained unstable, particularly those which were weakly unstable for . The new bifurcation diagram shown in Figure 10 is focused on the stable steady solutions. In order to have a clearer diagram, we completely suppressed all the unstable solutions and we also changed the variable plotted along the ordinate. It is now the Nusselt number which expresses the actual heat transfer through horizontal planes compared to the diffusive heat transfer and can be written simply in our case as . The choice of the Nusselt number can also give interesting information on how heat is transferred by the different solutions. The different stable branches begin at steady bifurcation points shown as solid circles and they terminate at oscillatory (Hopf) bifurcation points shown as open circles. The characteristics of these bifurcation points are given in Table 3. Although the bifurcation diagrams in Figures 5 and 10 look different because different characteristics are plotted, we want to recall that they correspond to the same bifurcation diagram. In particular we can verify that the values corresponding to the beginning of the stable solutions are the same in both diagrams. Note finally that the angular frequency given at a Hopf bifurcation point corresponds to the imaginary part of the corresponding critical eigenvalue. This frequency will be that of the oscillatory solution generated by the critical eigenvector very close to the Hopf bifurcation point. We now analyze the different stable branches.
Table 3. Existence Range of the Stable Steady Solutionsa
The stable steady solutions are from a steady bifurcation point to an oscillatory bifurcation point. Angular frequency and frequency of the oscillatory modes which eventually destabilize these solutions at the oscillatory bifurcation points.
 1. The stable one-roll branch is initiated at the first primary bifurcation point ( ) and terminates at a Hopf bifurcation point at . The frequency of the destabilizing oscillatory mode is . This stable branch is the first to appear, but also the first to disappear. For comparison, this oscillatory transition for the one-roll solution is given by Caltagirone  at and by Kimura et al.  at . The critical frequency estimated from a graph in the paper of Kimura et al.  is , a value a little strong compared to our value.
 2. The stable two-roll branch is a much longer branch. It is initiated at where it is stabilized and terminates at . The frequency of the destabilizing oscillatory mode is larger and equal to .
 3. The stable three-roll branch is initiated at where it is stabilized and terminates at . The frequency of the destabilizing oscillatory mode is .
 4. The stable four-roll branch is the last to appear, as this branch is only stabilized at , but it remains stable in a large range, up to a Hopf bifurcation point at . The frequency of the destabilizing oscillatory mode is also very large and corresponds to .
 From Figure 10, we can note that as the number of rolls of the solutions increases (from 1 to 4), the corresponding stable branches appear later, are destabilized later and exist over a larger range. The variations of the curves indicate that the Nusselt number increases with the increase of for each flow structure, indicating a better heat transfer connected to larger velocities. Moreover, when a new flow structure is stabilized, it appears with a Nusselt number smaller than the Nusselt number of the previously stabilized flow structures, but with the increase of , this new flow structure will eventually have the strongest Nusselt number. As a result, each flow structure induces the best heat transfer in a certain range of , successively the one-, two-, three-, and four-roll structures. This also indicates that the multiroll flow structures are more efficient for transferring heat when is increased. Nothing, however, indicates that the solution with the largest Nusselt number will be the preferred solution in this system.
 The existence range of all these stable steady solutions is summarized in Table 3. We better understand why, by time-evolution calculation, we can have only the one-roll solution for , the one-roll and two-roll solutions for , and the two-roll, three-roll, and four-roll solutions for . These stable steady solutions taken at the upper limit of their stable branches (Hopf bifurcation points) are shown in Figure 11. We see that the four solutions (from (a) with one roll to (d) with four rolls) look similar to those shown in Figure 8 for , except that the temperature field is more distorted by the stronger velocities and that boundary layers develop along the boundaries, principally the two horizontal boundaries.
4.4. Oscillatory Perturbations Destabilizing the Stable Steady Solutions
Figure 12 gives the complex eigenvectors (real and imaginary parts) which eventually destabilize the different stable steady branches at the Hopf bifurcation points. We recall that the perturbation P created by such complex eigenvectors (denoted as , where Hr is the real part and Hi is the imaginary part) is, close to the threshold, given by , where is the angular frequency given by the critical eigenvalue at the Hopf bifurcation point and denotes the real part. We can also write P as , which indicates that at time t = 0 (to within a translation in time) the perturbation is , at time (where T is the period) we have , at time we have , and at time we have .
 For the one-roll branch (Figure 12a), the perturbations occur along all the boundaries with maxima close to the upper-left and lower-right corners. The spatial shift between Hr and Hi in this case indicates a traveling wave behavior of the perturbations along the boundaries. Note that these perturbations break the central symmetry of the one-roll solution, which corresponds for the eigenvector to and .
 For the two-roll branch (Figure 12b), the perturbations are present along the ascending jet near the V plane and along the descending jets close to the right and left boundaries. More precisely, perturbations develop on the bottom wall on both sides of the V plane and, driven by the ascending jet, they merge to give some weaker, but larger-scale perturbations along the V plane. Similarly, perturbations develop on the top wall near the corners of the cavity and are transported by the descending jets along the lateral boundaries. All these perturbations keep the left-right symmetry of the two-roll solution.
 For the three-roll branch (Figure 12c), the perturbations develop very locally on the top and bottom walls, at the places where the descending and ascending jets are initiated. There are no perturbations along the lateral walls. The central symmetry SC is kept by the perturbations in this case.
 For the four-roll branch, there are different eigenvectors which have close thresholds. As for the three-roll branch, the perturbations for the critical eigenvector (Figure 12d) are localized on the top and bottom walls, in the three zones where the descending or ascending jets are initiated. These perturbations break the left-right SV symmetry of the four-roll branch. If the perturbations for the third eigenvector look similar to those just described, those associated with the second eigenvector are different. Only two zones of perturbations appear on the top wall, associated with the descending jets, and these perturbations keep the left-right SV symmetry.
 In section 4.5, the oscillatory solutions generated beyond the Hopf bifurcation points by these complex eigenvectors will be described.
4.5. Oscillatory Solutions
 Time-evolution calculations with the third-order time scheme have been performed close to the four Hopf bifurcation points which destabilize the different stable steady solutions. The time series and power spectra obtained in these four cases are shown in Figure 13. In each case, the solution at the Hopf bifurcation point is perturbed by adding the real part of the unstable complex eigenvector (see Figure 12a–12d) with a multiplicative constant. This constant is chosen so that the maximum vertical velocity of the perturbation is one-tenth of the maximum vertical velocity of the solution. The perturbed solution is then used to start the time evolution and the calculations are performed until a dimensionless time equal to 2 is reached. The variable plotted along the ordinate of the time series is the vertical velocity at a fixed location.
 1. The first time series (Figure 13a) is computed at , just above the Hopf bifurcation on the one-roll branch ( ). We obtain a very regular periodic time evolution which corresponds to a power spectrum dominated by a peak at , i.e., very close to the frequency of the critical eigenvector at the Hopf bifurcation point. We have checked that there is no oscillatory solution below and that the amplitude increases above . All this indicates that the Hopf bifurcation in this case is supercritical.
 2. The second time series (Figure 13b) is computed at , just above the corresponding Hopf bifurcation at on the two-roll branch. This time series seems to still correspond to a regular oscillation. In fact, the corresponding power spectrum presents a main peak at (i.e., very close to the frequency of the critical eigenvector at the Hopf bifurcation point), but also a small peak at which indicates a slight long wave modulation of the signal (two peaks also appear at ). In this case too, the Hopf bifurcation appears to be supercritical.
 3. The third time series (Figure 13c) is computed at , a little below the corresponding Hopf bifurcation at on the three-roll branch, because in this case the Hopf bifurcation was found to be subcritical. This subcritical character of the Hopf bifurcation means that a branch of unstable periodic limit cycles is to appear subcritically at the Hopf bifurcation point. The branch generally reverses direction at a saddle-node point where the limit cycles could be stabilized, and the limit cycles further evolve for increasing . At , the time series is still quite regular and the corresponding power spectrum presents a main peak at (i.e., close to the frequency of the critical eigenvector at the Hopf bifurcation point). Other smaller peaks are found at and which indicate a long wave modulation of the signal. This signal evolves very quickly for larger . Indeed, the modulation which is small for becomes important for , and at the signal already becomes chaotic.
 4. The fourth time series (Figure 13d) is computed at , just above the corresponding Hopf bifurcation at on the four-roll branch. This time series is now completely chaotic, and the corresponding power spectrum has not well defined peaks and rather appears as a continuous spectrum. For this case also, similar time series can be obtained below the critical Hopf bifurcation point, which indicates that the bifurcation is subcritical. In this case, however, the branch of limit cycles must undergo different oscillatory bifurcations along its path, which introduce new frequencies and lead to the observed chaotic behavior.
 Finally, for the three oscillatory solutions which appear as periodic, we have stored different snapshots during one period in order to better understand the oscillatory flow structure. These snapshots have been taken at regularly spaced times during the period (t = 0, , , , , and ). The corresponding plots showing the time evolution of the velocity or temperature fields are given for the one-roll solution in Figures 14 and 15, for the two-roll solution in Figure 16, and for the three-roll solution in Figure 17.
 For the one-roll solution, as the Hopf bifurcation is supercritical, the amplitudes of the oscillations close above this Hopf bifurcation point are weak. To have stronger oscillations which can be visible on the temperature and velocity fields, we have chosen stronger values of the Rayleigh number, namely and . For (Figure 14), we see traveling waves which move anticlockwise (i.e., in the direction of the roll) along the boundaries of the cavity. For example, the effect of the wave along the upper boundary can be well observed through the evolution of the temperature isolines deformations from t = 0 to , and similarly for the lower boundary from to (mod T). In fact, as the central symmetry SC of the one-roll solution (denoted as S) is broken by the eigenvector (denoted as H) at the Hopf bifurcation point, we have (symmetry of the solution) and (antisymmetry of the eigenvector). Close to the threshold, the oscillatory flow O can be written as where P is the oscillatory perturbation and a small amplitude. We then have , which gives . This explains why the oscillatory flow, which has lost the SC symmetry of the steady solution, presents a symmetry SC between states distant by half a period. The result obtained for (with the same initialization at the Hopf bifurcation point) is a little different (Figure 15) because the oscillatory flow has now the central symmetry SC. As for , anticlockwise traveling waves are found, but, because of the SC symmetry, the waves on the upper and lower boundaries now progress simultaneously. The results obtained for (with a period ) look similar to those obtained by Kimura et al.  for (see their Figure 3). The same traveling wave behavior with SC symmetry is shown, and the period is given as . Similar characteristics of the oscillatory flow are also found in the paper of Caltagirone [1975, Figure 6] for in a cavity with an aspect ratio (width/height) equal to 0.8.
 For the two-roll solution, the snapshots have been obtained for . We have chosen this value of because the flow is still almost periodic, but, as this value is close to the threshold, the perturbation amplitudes will remain small. The oscillatory solution is presented in Figure 16 in a single picture giving the time evolution of the temperature isolines. We see that the oscillatory perturbations mainly affect the ascending central jet and the descending lateral jets, as predicted by the structure of the critical eigenvector (Figure 12b). The main changes can be seen on the isolines −0.15 and 0.15 which are in zones of small temperature gradients. We see the detachment of a temperature bubble which occurs for the central jet between and and for the lateral jets between t = 0 and , i.e., with a phase shift of . This is in agreement with the eigenvector which has same sign perturbations for the ascending and descending jets. Indeed, a positive perturbation in the ascending jet will give a similar perturbation effect as a negative perturbation in the descending jet, but this negative perturbation will be obtained half a period later.
 For the three-roll solution, the snapshots have been obtained for and the results are presented in Figure 17 in a single picture as for the two-roll solution. At this value of and for the evolution time chosen, the flow is still almost periodic, but the perturbation amplitudes are small, which means that the subcritical character of the Hopf bifurcation must be rather small. The perturbations affect the basis of the ascending jet and the descending jet in the core of the cavity in a symmetric way, as predicted by the structure of the critical eigenvector (Figure 12c), but they also affect the two jets along the vertical boundaries.
 Finally, for the four-roll solution, the observed chaotic time behavior supports the occurrence of strongly perturbed flow structures. The main perturbations could be those connected to the dominant eigenvectors close to the Hopf bifurcation point or other perturbations which would have affected the limit cycle during its subcritical path.
5. Conclusions and Discussion
 The convection induced by solute effects in a porous square cavity has been investigated numerically using a spectral element code with bifurcation tools. The first six primary bifurcation points corresponding to the onset of convection from the trivial no-flow solution have first been determined. The complex bifurcation diagram initiated from these six points has then been calculated. It is found that four of these six branches corresponding to multiroll structures are eventually stabilized, the one-roll branch at the first primary bifurcation point at and the other branches later at secondary bifurcation points, the two-roll branch at , the three-roll branch at , and the four-roll branch at . These different values for the stabilization of the branches explain why, for example, only two different solutions can be obtained by time-evolution calculations for , and three for . Note that the stabilization process of the branches becomes more complex as the branch appears later, because the branch is multiply unstable at onset (e.g., four times for the four-roll branch). Some of these stabilization processes connected to interactions between modes are described in the paper of Riley and Winters .
 The stable branches have then been followed up to limit points where they eventually become unstable. These points which are always Hopf bifurcation points correspond to values which strongly increase as the roll number of the flow-structure increases. Except for the one-roll branch, these values are difficult to determine as they strongly depend on the precision of the mesh used, and meshes with up to points have been necessary to get accurate values. The one-roll branch is then stable up to , the two-roll branch up to , the three-roll branch up to and the four-roll branch up to . This explains also why for or , the one-roll steady solution cannot any more be obtained by time-evolution calculations.
 Time stepping calculations have then been performed to get the oscillatory solutions beyond the different Hopf bifurcation points. For the one-roll branch, the Hopf bifurcation appears to be supercritical and periodic oscillatory solutions can be obtained above the corresponding threshold. Close to the threshold, the oscillatory solution has the symmetries deduced from the steady solution and the corresponding eigenvector, but for larger an oscillatory solution with different symmetries is obtained. For the two-roll branch, the Hopf bifurcation also appears to be supercritical, but the oscillatory solution becomes nonperiodic (appearance of a second frequency) very close to the threshold. The Hopf bifurcations on the three-roll and four-roll branches, in contrast, appear to be subcritical. The oscillatory solution in the vicinity of the threshold is, however, almost periodic for the three-roll flow, whereas it is already chaotic for the four-roll flow.
 Our study thus indicates that below , only steady solutions with one, two, three or four rolls can be obtained. Beyond this value, a periodic oscillatory one-roll solution can be obtained, together with steady multiroll solutions. Two-roll oscillatory solutions can only appear beyond and they will quickly involve quasiperiodic oscillations. Three-roll and four-roll oscillatory solutions can eventually be triggered at still larger values, and for the four-roll solution, the flow will directly appear as chaotic.
 It is more difficult to predict which flow structure will be obtained during a real experiment, or during a given time-evolution simulation, as it will depend on the initial conditions, the rate of heating, the perturbations applied, and we have no numerical methods able to give these predictions in such complex systems. Indeed, the preferred solution will be the one-roll solution, as this solution is directly stable at its onset from the no-flow solution (at the first primary bifurcation), whereas the other solutions, sometimes called anomalous solutions [Riley and Winters, 1989], are stabilized at secondary bifurcation points. This one-roll solution is for example expected if the temperature difference is slowly increased (slow increase of the Rayleigh number). More precisely, under such conditions, the system is expected to follow the stable one-roll branch from its onset to the Hopf bifurcation point, and to give rise further to oscillatory one-roll flows. In contrast, the other solutions (multiroll solutions) will benefit from abrupt changes and irregularities (strong heating, imperfect boundary conditions, presence of perturbations). Once such multiroll solutions have been reached, the corresponding stable branches can be followed for regularly increasing , and even oscillatory flows of the same type can be further triggered. Despite the fact that these multiroll solutions correspond to a better heat transfer at large , it is not expected that they will be automatically the preferred solution because no natural mechanism has been found to allow a jump to their corresponding branches.
 All the results obtained in this study have a little clarified the existence of multiple flow solutions in the Horton-Rogers-Lapwood problem. They also can shed light on more complex and more real situations, as those concerned with density-dependent groundwater flows. They can finally be used as simple test cases.
 This work was granted access to the HPC resources of IDRIS under the allocation 2011-021559 made by GENCI (Grand Equipement National de Calcul Intensif).