## Abstract

This paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 : 2. A two-layered Rayleigh–Bénard problem with infinite extent is examined; two layers are separated by a non-deformable and thin dividing plate. This physical set-up is known to provide an exact 1 : 2 resonance between critical modes. Restricting the wavevectors of interacting modes on a hexagonal lattice, we derive 12-dimensional amplitude equations up to cubic order. Among steady solutions of the equations, the stable ones are hexagons and mixed hexagons. Travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values.

## 1. Introduction

Spatial resonance between steady modes, whose wavenumbers are in the ratio 1 : 2, has been investigated both from a dynamical systems point of view and from a physics point of view (Busse & Or 1986; Dangelmayr 1986; Armbruster 1987; Buzano & Russo 1987; Armbruster *et al*. 1988; Proctor & Jones 1988; Manogg & Metzener 1994; Mercader *et al*. 2001, 2002*a*,*b*; Porter & Knobloch 2001, 2005; Nore *et al*. 2003, 2005, among others). Under O(2)-symmetry, amplitude equations for the 1 : 2 resonance are known to possess two kinds of steady solutions, namely pure modes and mixed modes. Travelling waves bifurcate from the mixed modes as relative equilibria. As Hopf bifurcations, standing waves bifurcate off the mixed modes and modulated waves bifurcate off the travelling waves. (See Dangelmayr (1986) for example.) Armbruster *et al*. (1988) and Proctor & Jones (1988) found structurally stable heteroclinic cycles. The heteroclinic cycles are robust and are found in nonlinear partial differential equations (PDEs) governing the Rayleigh–Bénard problem (Mercader *et al*. 2002*a*,*b*) and in an experiment on von Kármán swirling flow (Nore *et al*. 2003, 2005). Moreover, far above the mode interaction point or under a weak break of a reflectional symmetry, Porter & Knobloch (2001, 2005) found new complex dynamics. So far, all the results above exhibit two-dimensional spatial structures that vary periodically in a horizontal direction. Hereafter, such spatial structures will be referred to as two dimensional. A question naturally arises as to whether or not the bifurcation characteristics in two-dimensional problems still keep their physical significance in three-dimensional problems.

In the presence of the 1 : 2 resonance, on a two-dimensional wavenumber plane, there are two critical circles whose radii are in the ratio 1 : 2. Since the wavevectors of interacting modes are densely distributed on the circles, it is difficult to examine the interaction without any simplification. In this paper, we simplify the situation by restricting the wavevectors to lie on a hexagonal lattice in the wavenumber plane; we consider a case where exactly six lattice points are on the critical circle with the radius *k*_{c} and six points are on the circle with the radius 2*k*_{c}.

Pattern formation on a hexagonal lattice has a long history of investigations since the late 1950s; special emphasis is on selection mechanisms between rolls and hexagonal cells in the Rayleigh–Bénard (hereinafter referred to as RB) problem and the Bénard–Marangoni problem. See Busse (1978, 1989) and references therein. Systematic analyses based on equivariant bifurcation theory have been carried out by Buzano & Golubitsky (1983) and Golubitsky *et al*. (1984) for steady modes. It is well established that in the absence of up–down symmetry in the horizontal mid-plane, generic amplitude equations possess two primary solutions exhibiting rolls and hexagons. Solutions exhibiting rectangles and triangles may bifurcate from them as the secondary solutions. Dionne *et al*. (1997) extended the theory by assuming that points of a finer square lattice or points of a finer hexagonal lattice are on the critical circle with the radius *k*_{c}; they showed a rich variety of planform involving superlattice patterns. See also Silber & Proctor (1998), Skeldon & Silber (1998), and Judd & Silber (2000). For the Hopf onset, see Roberts *et al*. (1986) and Renardy & Renardy (1988).

Mode interactions on a hexagonal lattice have also been examined during the last decade. For example, Daumont *et al*. (1997) investigated the interaction between modes with wavenumbers in the ratio : 1 and reported a drifting pattern and an oscillatory hexagonal pattern caused by a break of symmetry.

Beyond the scope of the present paper is pattern formation on a square lattice: see Clune & Knobloch (1994) and Silber & Knobloch (1991), respectively, for steady onset and Hopf onset. Dawes (2001) investigated a Hopf bifurcation on a finer square lattice. For mode interactions on a square lattice, Proctor & Matthews (1996) and Podvigina & Ashwin (2007) studied the : 1 resonance between steady modes and Dawes (2000) analysed the : 1 steady/Hopf mode interaction.

Both the : 1 resonance and the : 1 resonances in the three-dimensional problem are strong in the sense that a phase coupling between interacting modes is at the quadratic order. In the two-dimensional problem, on the other hand, a phase coupling between modes with wavenumbers in the ratio : 1 or : 1 is entirely absent; exchange of energy between interacting modes is only due to the moduli of complex amplitudes. Therefore, the interactions are non-resonant.

The objective of the present paper is to examine the effect of the 1 : 2 resonance in the three-dimensional pattern formation problem. Let us focus on an RB problem even though no experimental results are available on three-dimensional patterns which are caused by the 1 : 2 resonance. A pure RB problem with a single layer of fluid, which is bounded by two rigid plates, possesses a codimension two point on the neutral curve at . At this point, the 1 : 2 resonance occurs between non-critical neutral modes. In the two-dimensional problem, where all the wavevectors are aligned in a horizontal direction, we may excite resonating modes artificially by applying a thermal imprinting technique (Chen & Whitehead 1968). Therefore, the resonance between non-critical neutral modes is considered to be physically achievable. In the three-dimensional problem, a skilful excitation of resonating modes should be introduced to observe convection pattern caused by the resonance. In a two-layered RB problem, on the other hand, Proctor & Jones (1988) found that the 1 : 2 resonance may set in between critical modes. The two- layered problem thus provides an ideal example of the exact 1 : 2 resonance that may generate a two-dimensional planform in a horizontal plane with infinite extent. On a hexagonal lattice, amplitude equations involve two kinds of quadratic nonlinear terms: the first is due to the 1 : 2 resonance and the second is due to the quadratic interaction between modes on the same critical circle.

This paper is organized as follows. In §2, we will briefly describe the physical set-up of the two-layered RB problem, the mathematical formulation and the linear stability characteristics. Since the problem is borrowed from Proctor & Jones' paper, we suppress the details as much as we can. The steady solutions of the amplitude equations will be partly classified in §3. In §4, we will examine the bifurcation of steady solutions as well as travelling wave solutions for specific parameter sets. Some concluding remarks will be given in §5.

Since we consider the 1 : 2 resonance in spatially extended systems where the periodicity is chosen in accordance with the minima of the neutral curve, the problem is of codimension three. Indeed, criticality, equal heights of minima and resonance require to adjust three parameters. Then, allowing generic variations around such a codimension three point, the resonance will also be broken and one may expect commensurate–incommensurate transitions. Since we choose parameters to preserve the resonance, the present analysis does not capture all generic perturbations.

## 2. Two-layered Rayleigh–Bénard problem

### (a) Physical set-up and mathematical formulation

Proctor & Jones (1988) proposed an ideal problem of convection which allows an exact 1 : 2 resonance between critical modes. Following Proctor & Jones, we consider an RB problem composed of two horizontal fluid layers with infinite extent. The layers are contained between the top and the bottom horizontal plates and a horizontal ‘dividing plate’ which is non-deformable, heat conducting, heated and thin. The bottom plate is heated and the top plate is cooled at different but uniform temperatures. The dividing plate avoids the mechanical coupling between the top and the bottom layers. The thermal coupling, on the other hand, may allow the exact 1 : 2 resonance between the critical modes. In order for this paper to be self-contained, we briefly describe the physical set-up and the mathematical formulation of the problem. See Proctor & Jones (1988) for further details.

We assume that the Boussinesq approximation holds; only the densities in the buoyancy terms are functions of the temperature. As has been pointed out by Schlüter *et al*. (1965), amplitude equations of the cubic order are not generic if the linear operators involved in the governing equations are self-adjoint. To avoid a non-generic situation, we also assume that the densities in the buoyancy terms are weak quadratic functions of the temperature; the quadratic density profiles guarantee that the linear operators are non-self-adjoint. The latter assumption is the only difference between our problem and the problem in Proctor & Jones.

Assume that the bottom and the top plates are located at *z*^{*}=0 and *d*(1+*D*^{−1}), respectively, and the dividing plate is at *z*^{*}=*d*. The temperature on the bottom plate is maintained at *T*^{*}=*T*_{b}, whereas the temperature on the top is at *T*_{t}. The temperature on the dividing plate is *T*^{*}=*T*_{d}.

We attach suffixes 1 and 2 to indicate the variables and the physical properties in the lower and upper layers, respectively. The velocities and , the pressures and and the temperatures and are governed by(2.1)where, *e*_{z} is the unit vector upward in the *z*-direction; *g* is the acceleration due to the gravity; *μ*_{1} and *μ*_{2} are the viscous coefficients; *κ*_{1} and *κ*_{2} are the thermal diffusivities; and and are the densities. These physical properties are evaluated at and .

Let us now non-dimensionalize (2.1) by setting(2.2)where we set , and .

Following Proctor & Jones, we assume *κ*_{1}=*κ*_{2}. We further set . In the non-dimensional form, the equations governing the disturbance are written as(2.3)where . Equations (2.3) involve non-dimensional parameters defined by(2.4)The *ϵ*_{j} measure the effect of the quadratic deformation from the linear density profile.

We impose the boundary conditions(2.5)On the dividing plate, the temperatures need to satisfy(2.6)which yield(2.7)

We eliminate the pressure terms from (2.3) by taking (*∇*×) and (*∇*×)^{2} of the equations for *v*_{j} and linearize the resultant equations.

Let us now introduce the normal mode(2.8)We then obtain a linear eigenvalue problem of the form with appropriate linear operators *S* and *L* under the homogeneous boundary conditions

(2.9)

The adjoint problem is defined by , where for *j*=1,2. The 〈,〉 denotes an appropriate inner product.

### (b) Linear stability characteristics

We solve the linear eigenvalue problem and the adjoint problem by expanding in Chebyshev polynomials. The boundary conditions at *z*=1 are imposed by means of the Tau method. An application of the collocation method yields an algebraic eigenvalue problem. The QZ package of the IMSL is used to solve the problem, numerically. In figure 1, we show the linear neutral curves for several sets of *ϵ*_{1} and *ϵ*_{2} with appropriate values of . We fixed the depth ratio at *D*=2.0977. The neutral curves exhibit the exact 1 : 2 resonance, i.e. the wavenumbers of two local minima are *k*_{c} and 2*k*_{c} and the corresponding Rayleigh numbers are the same. We confirmed the accuracy of the parameter values (table 1 of Proctor & Jones) which give exact resonance, i.e. *R*_{1}=1401.8, and *k*_{c}=2.9150 for *D*=2.0977 under the linear density profiles *ϵ*_{1}=*ϵ*_{2}=0. As long as we examined, the perturbation from (*ϵ*_{1}, *ϵ*_{2})=(0, 0) to (0.5, 0.5), (0, 0.5) or (0.5, 0) preserved the resonance. For those perturbations, with fixed , we can adjust to preserve the resonance. In this paper, to avoid tremendous parameter survey, we assumed that at least the top fluid layer is filled with water for which the density has the maximum at 4°C. Other than water, the quadratic correction on the density is weak at room temperature. We note that for all the cases in the figure.1

## 3. Steady solutions of the amplitude equations

Denote a disturbance by . Since we consider the 1 : 2 resonance on a hexagonal lattice, the is assumed to have the form(3.1)where denotes the eigenfunction belonging to the first eigenvalue.

Let the amplitude equations(3.2)for be generated by the vector field(3.3)The vector field is said to be equivariant under an action of a symmetry group *Γ* if holds for all *γ*∈*Γ*. A hexagonal lattice has the group of symmetry , where D_{6} is the dihedral group of the order of 12 and T^{2} is the two-dimensional torus on a plane. The dihedral group D_{6} is generated by the inversion through the origin *c* and D_{3}. The *c* acts on ^{6} as . The D_{3} is generated by the anticlockwise rotation by the angle 2π/3, *R*_{2π/3}, acting on ^{6} as , and the reflection in a vertical plane, *σ*_{v}, acting on ^{6} as . The acts on ^{6} as for . See Buzano & Golubitsky (1983), Golubitsky *et al*. (1984), or Golubitsky *et al*. (1988) for further details.

The determination of the vector field that is -equivariant is not easy although it is standard. In appendix A of the electronic supplementary material, we give a preliminary result.

The Taylor expansion of the resultant about the origin and the truncation at the cubic order yield the amplitude equations(3.4)and

We assume the linear operators in (2.3) being non-self-adjoint so that *δ*_{1} and *δ*_{2} do not vanish. Solutions of the amplitude equations (3.2) are written as(3.5)We set , and for *l*=1,2 and 3.

In the remaining part of this section, we list steady solutions of with , labelled by S_{1}–S_{10}, together with higher dimensional ones S_{11}–S_{13} arising in our two-layered problem. Here, denotes the fixed-point subspace of a subgroup defined by . The eigenvalues of the Jacobian matrix about the steady solutions S_{1}–S_{9} are given in appendix B of the electronic supplementary material.

S_{1}. . It corresponds to the conduction state having the full symmetry .

S_{2}. for *x*∈. This solution exhibits pure rolls with the wavelength *π*/*k*_{c}. In figure 2, the convection pattern of it together with other typical ones are demonstrated in the square and . The isotropy subgroup of a solution *z*_{0} that is defined by is . The *x* and *σ*_{2} are related by .

S_{3}. for *x*∈. For this solution, we require that *z*_{1}=*z*_{2}=*z*_{3}=0, *z*_{4}=*z*_{5}=*z*_{6}≠0 and *Φ*_{2}=*nπ* for an integer *n*. An even *n* gives positive *x* within a phase shift. By S_{3}^{+}, we denote the S_{3} with a positive *x*. It exhibits up-hexagons (figure 2). An odd *n*, on the other hand, gives negative *x*. By S_{3}^{−}, we denote the S_{3} with a negative *x*. It exhibits down-hexagons. . Based on the amplitude equations of the cubic order, the *x* and *σ*_{2} are related by . In the *σ*_{1}*σ*_{2}-plane, the existence of the solution S_{3} is bounded by the line .

We consider the generic situation under the symmetry of a hexagonal lattice, . This means there exists no solution satisfying and . Among solutions of (3.4) satisfying and , there are only three solutions for which isotropy subgroups are axial; they are S_{2}, S_{3}^{+} and S_{3}^{−} as was shown by Buzano & Golubitsky (1983). Their solution branches bifurcate from the conduction state on the line *σ*_{2}=0.

For , , but with , see the solution S_{7} below.

S_{4}. for . Here, we require that for an integer *n*. This solution exhibits mixed rolls with the wavelengths *π*/*k*_{c} and 2*π*/*k*_{c} (figure 2). . The *x* and *y* satisfy the equations and . In the *σ*_{1}*σ*_{2}-plane, the solution branch of S_{4} bifurcates from the conduction state on the line *σ*_{1}=0 and from the branch of S_{2} on the line given by and .

For , we have a relative equilibrium which exhibits travelling waves. This case will be discussed further in §4. For relative equilibria, see Golubitsky & Stewart (2002) for example.

S_{5}. for with . This solution exhibits rectangles (or false hexagons) with the wavelength *π*/*k*_{c}; the solution branch of S_{5} bridges between the branches of S_{2} and S_{3}. . The *x* and *y* satisfy the equations and .

S_{6}. for . For this solution, we require and implying that and . If we assume and or and for integers *l*, *m* and *n*, we respectively have or , and the resultant planform exhibits mixed hexagons with the wavelengths *π*/*k*_{c} and 2*π*/*k*_{c}. If we assume and , instead, we have and . The formed pattern is the same as the mixed hexagons given by with , within a phase shift. The signs of *x* and *y* correspond to the signs of cos *mπ* and cos *nπ*, respectively. Depending on the signs of *x* and *y*, there are four variations as are shown in figure 2: S_{6}^{++} for ; S_{6}^{+−} for *x*>0 and *y*<0; S_{6}^{−+} for *x*<0 and *y*>0; and S_{6}^{−−} for . D_{6}. The *x* and *y* are real solutions of the branching equations and . In the *σ*_{1}*σ*_{2}-plane, the solution branch of S_{6} bifurcates from the conduction state on the line *σ*_{1}=0 and from the branch of S_{3} on the line given by and .

If and , we have since . See the solution S_{11}.

S_{7}. for . This solution exhibits triangles with the wavelength *π*/*k*_{c}, . Note that the solution S_{7} occurs as a solution of (3.4) if nonlinear terms of the quartic order or higher are involved in (3.4).

S_{8}. for . This solution exhibits a mixture of rolls with the wavelength 2*π*/*k*_{c} and rectangles with the wavelength *π*/*k*_{c}. The solution branch of S_{8} bridges between the branches of S_{4} and S_{3} or between those of S_{4} and S_{5}. . The branch of S_{8} bifurcates from the conduction state at the origin and from the branch of S_{3} on the line given by and .

S_{9}. for . .

S_{10}. for and . .

S_{11}. for .

This solution exhibits mixed triangles with the wavelengths *π*/*k*_{c} and 2*π*/*k*_{c}, D_{3}.

S_{12}. for where and . . The solution S_{12} exhibits mixed rectangles with the wavelengths *π*/*k*_{c} and 2*π*/*k*_{c}. The branch of S_{12} bifurcates from the branch of S_{2} on the line .

S_{13}. for where . . The branch of S_{13} bifurcates from the branch of S_{2} on the line .

## 4. Numerical results

By means of expansions in double Fourier series and linear eigenfunctions, we reduced the nonlinear PDEs (2.3) to infinite-dimensional ordinary differential equations (ODEs). By carrying out the centre manifold projection (Carr 1981), we derived the amplitude equations (3.4) from the infinite-dimensional ODEs. Details of the reduction are given in the appendix C of the electronic supplementary material. In the course of our numerical evaluation of the coefficients, we normalized the eigenfunctions so as to satisfy . The adjoint function was normalized as if and otherwise, where . We evaluated all the coefficients in (3.4) for both a one-fluid model with and and a two-fluid model with *G*=1, and . In the former, we assumed that , whereas in the latter we assumed that and . In table 1, we tabulated the coefficients in typical cases for . In the evaluation, we truncated the number of eigenfunctions at 20.2 This truncation is consistent with that for a single-layer RB problem (Fujimura 1997). We need at least 49 Fourier modes to derive the amplitude equations (3.4). Our centre manifold reduction is thus from the 20×49-dimensional system to the 12-dimensional one.

For , the linear operators of our problem are self-adjoint so that *δ*_{1} and *δ*_{2} vanish as had been pointed out by Schlüter *et al*. (1965). In the absence of the 1 : 2 resonance, we need to proceed to the quartic-order approximation to distinguish between up- and down-hexagons, and to the quintic order to specify triangles. To avoid tremendously heavy manipulations in deriving the amplitude equations of the higher order, we have weakly perturbed the density profile by adding a weak quadratic term (see (2.1)). We may find from table 1 how the *δ*_{1} and *δ*_{1} recover non-vanishing values when *ϵ*_{1} and *ϵ*_{2} deviate from (0, 0). A slight increase of *ϵ*_{2} causes a significant effect on the non-self-adjointness. Suppose that °C and °C. Then, the *ϵ*_{1} and *ϵ*_{2} are of the order of 10^{−1} for water (*P*=7). Therefore, the values of *ϵ*_{1} and *ϵ*_{2} in table 1 are easily achieved in laboratory experiments.

Let us now examine the steady solutions only for one set of parameters in table 1, namely *P*_{1}=*P*_{2}=7, *D*=2.0977, *ϵ*_{1}=*ϵ*_{2}=0.1 and . First, we control the *σ*_{1} and *σ*_{2} by changing *R*_{1} near the critical value *R*_{1c} while keeping fixed at 1.2180.

Figure 3 shows bifurcation diagrams exhibiting steady solution branches of (3.4) with relatively low dim Fix. By restricting the solution in an invariant subspace , we obtained branches of steady solutions as shown in figure 3*a*. Since the linear operators are not self-adjoint for , up-hexagons S_{3}^{+} and down-hexagons S_{3}^{−} are clearly distinguished. This diagram is consistent with that for the RB problem with a single layer, for which the Boussinesq approximation does not hold (Busse 1967). The 1 : 2 resonance changes the stability of the solutions S_{3}^{+} and S_{2}^{−} as is found in figure 3*b*, where the limitation on the invariant subspace is removed. Note that the solution branch of S_{8} together with solution branches of S_{2}, S_{3}^{+} and S_{3}^{−} bifurcate from the conduction state at . The stable solution S_{3}^{+} in figure 3*a* is now destabilized by the positive eigenvalue with multiplicity three. Here, and . The stable solution S_{2} in figure 3*a* is destabilized by the eigenvalues and with multiplicity four. These positive eigenvalues are due to the 1 : 2 resonance. Mixed hexagons S_{6}^{++} are found to be stable above the saddle-node point, instead.

To see the origin of the branches S_{6}, we set(4.1)and change *φ* with prescribed small *ρ*.

In figure 4*a*–*d*, branches of the steady solutions described in §3 are depicted for *ρ*=10^{−4}. The *ρ* is so small that, qualitatively, the bifurcation diagrams no longer change for even smaller *ρ*. For the orbital stability of the solution branches S_{2}–S_{6}, see appendix D of the electronic supplementary material.

As the primary solutions, both the branches of rolls S_{2} and hexagons S_{3}^{±} bifurcate off the trivial solution branch at and 2*π*. The solution S_{2} exists in ; the solutions S_{3}^{−} and S_{3}^{+} exists in and , respectively. Since figure 4 is for small solutions, the solution branch of S_{3}^{+} with relatively large norm is not shown although it does exist everywhere in . See figure 3 or 6*a* for further details.

In figure 4*a*, we show the bifurcation diagram by restricting in another invariant subspace . This subspace gives spatial patterns caused by the 1 : 2 resonance in the two-dimensional problem. A part of the branch S_{4} and a part of the branch TW_{1} are found to be stable. The label TW_{1} denotes the travelling waves briefly mentioned in §3. As a relative equilibrium, this solution is steady in terms of *r*_{1}, *r*_{4} and *Θ*_{1} with sin *Θ*_{1}≠0. Owing to non-vanishing sin *Θ*_{1}, both *θ*_{1} and *θ*_{4} are proportional to *t* sin *Θ*_{1}. The solution TW_{1} thus exhibits a travelling (or drifting) wave feature. In its planform, mixed rolls propagate perpendicular to their roll axes.

We now remove the limitation on the invariant subspace. Recall that, in figure 3*b*, the 1 : 2 resonance destabilizes both the rolls S_{2} and the up-hexagons S_{3}^{+}. The solution of mixed hexagons S_{6}^{++} is stabilized, instead. Figure 4*b* shows that, in addition to the solution TW_{1} in figure 4*a*, a branch of another relative equilibrium, TW_{2}, bridges between the branches of S_{6}^{+−} and S_{6}^{−+}. The solution TW_{2} is partly stable, whereas the solution TW_{1} is always unstable.

Consider now equation (3.5). For travelling waves TW_{2}, we require that *r*_{2}=*r*_{3}, *r*_{5}=*r*_{6}, *Φ*_{1}=const., *Φ*_{2}=const., *Θ*_{1}=const., *Θ*_{2}=const. and *Θ*_{3}=const. The are thus constant, implying that for constant and . Since *Θ*_{k}=const. for *k*=1, 2 and 3, we have and . Moreover, the equalities and yield and . In the 12-dimensional ODEs for *r*_{j} and *θ*_{j}, the equalities *r*_{2}=*r*_{3} and *r*_{5}=*r*_{6} imply as well as and . We thus have and . The travelling waves TW_{2} lie on the group orbit for . Setting and , we haveTherefore, the spatial pattern of TW_{2}, which is shown in figure 5*a*, propagates in the *x*-direction with the phase velocity *c*. For this specific example at , *c*<0.

Figure 4*c* is a magnification of the rectangular region in figure 4*b*. The tiny rectangular region in figure 4*c* is further magnified in figure 4*d*. For steady solutions, see §3.

Figure 6*a*–*d* show the solution branches S_{2}, S_{3}, S_{4}, S_{6}, TW_{1} and TW_{2} for four values of *ρ*. Even though the solutions are no longer local for *ρ*≥10^{−3}, we see how the solutions S_{6}^{++}, S_{6}^{−+} and S_{3}^{+} are stabilized with the increase of *ρ*. In figure 6*d*, the solution of travelling waves TW_{1} are stabilized within a narrow range close to .

Using bifurcation data like the ones in figure 6 for various *ρ*, stable regions of the solutions S_{3} and S_{6} are obtained in the *σ*_{1}*σ*_{2}-plane. Instead, figure 7 shows stable regions of S_{3} and S_{6} in the *rR*_{1}-plane. The solutions of mixed hexagons S_{6}^{++} and S_{6}^{+−} exist stably in the regions which cover wide areas in the *rR*_{1}-plane and spread out from the neighbourhood of the interaction point . We thus expect the mixed hexagons S_{6}^{++} and S_{6}^{+−} observable in laboratory experiments using water as working fluids. In contrast, the hexagons S_{3}^{+} and the mixed hexagons S_{6}^{−+} are difficult to realize owing to their narrow stable regions. Figure 7*b* is a magnification of figure 7*a* near the interaction point after an appropriate linear transformation of the *rR*_{1}-plane. Figure 6*b*–*d* as well as figure 7*a*,*b* involves non-local bifurcation characteristics. The validity of these figures can be confirmed only through the Euler–Newton continuation based on governing nonlinear PDEs (2.3) under (2.5) and (2.7).

## 5. Concluding remarks

We reported numerical results for a one-fluid model with *P*_{1}=*P*_{2}=7 and *ϵ*_{1}=*ϵ*_{2}=0.1. (Hereinafter referred to as (i).) Among steady solutions, only the solution of mixed hexagons S_{6}^{+−} is stable in a narrow range of parameters if . Relaxation of the restriction on *ρ* enables solutions of mixed hexagons S_{6}^{++} and S_{6}^{+−} to exist stably in much wider ranges. The solution of travelling waves TW_{2} may also exist stably as a relative equilibrium.

In addition to the one-fluid model (i) with *P*_{1}=*P*_{2}=7 and *ϵ*_{1}=*ϵ*_{2}=0.1, we also examined three other cases, namely (ii) a one-fluid model with *P*_{1}=*P*_{2}=7 and *ϵ*_{1}=*ϵ*_{2}=0.2, (iii) a two-fluid model with *P*_{1}=150.76, *P*_{2}=7, *ϵ*_{1}=0 and *ϵ*_{2}=0.1, and (iv) a two-fluid model with *P*_{1}=158.48, *P*_{2}=7, *ϵ*_{1}=0 and *ϵ*_{2}=0.2. The coefficients of the amplitude equations for these cases are listed in table 1.

In case (ii), the solutions S_{6}^{+−} and TW_{2} still possess wide stable regions, whereas the solution S_{6}^{++} is no longer stable. Instead, an oscillatory solution, OS say, may exist stably. It satisfies , , and , implying that . For a snapshot of this solution, see figure 5*b*. The equations governing the oscillatory solution OS are two-dimensional for :(5.1)Steady solutions of (5.1) include the solutions S_{1}, S_{3}^{+} and S_{6}^{++}. In the parameter range where the solution OS is stable, the 2×2 Jacobian matrix about the steady solution S_{6}^{++} has complex conjugate eigenvalues whose real parts are positive. A forward integration of (5.1) yields the oscillatory solution OS, whereas a backward integration yields the steady solution S_{6}^{++} under the same non-vanishing initial conditions and . As is guaranteed by the equations , the solution OS oscillates around the unstable steady solution S_{6}^{++}, keeping the *r*_{1} and *r*_{4} positive.

In the cases (iii) and (iv), the solution S_{3}^{+} has a wide stable region which is isolated from the interaction point. Stable regions of mixed hexagons and time-periodic solutions are too narrow to be observed in experiments. The solution of mixed triangles S_{11} is found to exist stably in a very narrow parameter region.

Busse & Sommermann (1996) observed travelling waves and standing waves in a two-layered RB experiment, where they did not insert a dividing plate to make the two-layered set-up. Instead, they used two immiscible liquids. In order to observe travelling waves caused by the 1 : 2 resonance, they made the experimental apparatus in an annular shape. In contrast, again using two immiscible liquids but with a straight channel, Andereck *et al*. (1996) observed, besides travelling waves, an oscillation between thermal coupling and mechanical coupling between two layers. If the fluid layers have sufficiently large extent in a horizontal plane, we expect that mixed hexagons S_{6}^{++}, S_{6}^{+−}, three-dimensional travelling waves TW_{2} or the three-dimensional oscillatory mode OS may be observed instead of two-dimensional travelling waves TW_{1}.

In two-dimensional 1 : 2 resonance under O(2)-symmetry, structurally stable heteroclinic cycles have been found by Armbruster *et al*. (1988), by Proctor & Jones (1988) in two-dimensional ODEs under O(2)-symmetry, by Mercader *et al*. (2002*b*) in PDEs governing two-dimensional RB convection and by Nore *et al*. (2003, 2005) in PDEs and laboratory experiments on von Kármán swirling flow.

We examined structurally stable heteroclinic cycles in the two-dimensional problem with respect to perturbations on a hexagonal lattice through numerical integrations of the amplitude equations (3.4). Restricting the solution in two-dimensional invariant subspace ( say), we found that nearly heteroclinic cycles existed in parameter ranges: for *ρ*=10^{−4} and 10^{−3}, for *ρ*=0.01 and for *ρ*=0.02 in case (i); for *ρ*=10^{−4}, for *ρ*=10^{−3}, for *ρ*=0.01 and for *ρ*=0.02 in case (ii); for *ρ*=10^{−4}, for *ρ*=10^{−3}, for *ρ*=0.01 and for *ρ*=0.02 in case (iii); and for *ρ*=10^{−4}, for *ρ*=10^{−3}, for *ρ*=0.01 and for *ρ*=0.02 in case (iv). We examined their stability with respect to perturbations in ^{6} simply by adding small perturbations of the order of 10^{−24} a long time after the establishment of the cycles. We found that in all the cases above, nearly heteroclinic cycles in *I*_{2} were unstable to small perturbations in ^{6}. Figure 8*a* shows an example of the breakdown of the cycles in *I*_{2} into new cycles in four-dimensional invariant subspace (). Nearly heteroclinic cycles in *I*_{4} exist in parameter ranges: for *ρ*=10^{−4} and for *ρ*=10^{−3} in case (iii) and for *ρ*=10^{−4}, for *ρ*=10^{−3}, for *ρ*=0.01 and for *ρ*=0.02 in case (iv). The cycles in *I*_{4} are stable to perturbations of *O*(10^{−24}) in ^{6} only for with *ρ*=10^{−3} in case (iii). For all the other parameter values that we examined, they are found to be unstable. We have to note, however, that with the increase of perturbations above *O*(10^{−17}), the stable region of nearly heteroclinic cycles in *I*_{4} shrinks, and vanishes for perturbations larger than of the order of 10^{−6}. Nearly heteroclinic cycles in *I*_{4} break down into a periodic or quasi-periodic state. An example of the breakdown of nearly heteroclinic cycles in *I*_{2} is shown in figure 8*b* where the cycle *I*_{2} is changed to a periodic solution in ^{6} via nearly heteroclinic cycles in *I*_{4} and almost chaotic state in ^{6}.

In a two-dimensional RB problem with a single layer, Busse & Or (1986) and Armbruster (1987) investigated the 1 : 2 interaction at the codimension two point on the neutral curve. Assume that a Boussinesq fluid has a linear density profile and is contained between two rigid plates. Then, an additional Z_{2}-symmetry needs to be imposed since governing equations have an up–down symmetry in the horizontal mid-plane. Generic amplitude equations need to involve nonlinear terms of the quintic order. Armbruster examined the effect of the quintic terms on the amplitude equations whose vector field is O(2)×Z_{2}-equivariant. Let us consider a situation where such a resonance takes place in the three-dimensional RB problem with a single layer. The Z_{2} acts on ^{6} as . Under , steady solutions with the axial subgroups involve rolls and , patchwork quilts and , hexagons and , and regular triangles and where . They are pure modes with wavenumber *k*_{c} and pure modes with 2*k*_{c}. We thus have a rich variety of possible convection patterns caused by the additional symmetry Z_{2}. A non-Boussinesq effect, a nonlinear density profile, or asymmetric boundary conditions breaks Z_{2} symmetry; in this case, the present analysis is valid. Busse & Sommermann (1996), however, pointed out ‘it seems unlikely that such an interaction phenomenon can be realized in a single-layer convection experiment.’

So far, we have investigated the 1 : 2 resonance on a hexagonal lattice. We have not considered the resonance on a square lattice. Suppose that the centre eigenspace is spanned by *ζ*_{1}, *ζ*_{2}, *ζ*_{3} and *ζ*_{4}, say, which means the disturbance having the form . In the absence of the additional symmetry Z_{2}, the *ζ*_{1} resonates with *ζ*_{3} and *ζ*_{2} resonates with *ζ*_{4} at the quadratic order. On the other hand, at the cubic order, the *ζ*_{1} interacts with *ζ*_{2} and *ζ*_{3} interacts with *ζ*_{4} non-resonantly through their moduli. Therefore, couplings between *ζ*_{1} and *ζ*_{3} and between *ζ*_{2} and *ζ*_{4} are much stronger than those between *ζ*_{1} and *ζ*_{2} and between *ζ*_{3} and *ζ*_{4}. We thus expect that, on a square lattice, two-dimensional patterns will be preferred to three-dimensional patterns. To conclude this, concrete analyses are needed.

In this paper, we have examined the bifurcation of solutions in (3.4). The relationship between the amplitude and the disturbance is given by (3.1) where the eigenfunctions *ϕ*_{1}, …, *ϕ*_{6} are normalized by as is described in the first paragraph of §4. In figure 9, we show the temperature field of the disturbance in a cross section of a roll-type structure. Figure 9*a* shows the temperature field for , whereas figure 9*b* shows the one for . Figure 9*c* shows a cross section of the solution S_{4}. For the solution S_{4}, *z*_{1} and *z*_{4} are typically 10^{−5} and 10^{−4}, respectively, for *ρ*=10^{−4} in case (iii). Figure 9*c* is thus due to the superposition of figure 9*a*,*b* after multiplying by 10^{−5} and 10^{−4}, respectively. Consider a situation in which thermo-sensitive liquid crystal powder is added to the working fluids. We may visualize a planform of convection pattern from above or below by applying a horizontal light sheet to the fluid layers. Since the planform depend on the height as is clearly seen in figure 9*c*, planforms like figure 2 can be observed when the height of the light sheet is appropriately adjusted.

## Acknowledgments

The author expresses his sincere thanks to referees for their helpful and constructive comments on the manuscript.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0168 or via http://www.journals.royalsoc.ac.uk.

↵Although we do not intend to examine the : 1 resonance, our numerical analysis shows the onset of the exact : 1 resonance. For

*ϵ*_{1}=*ϵ*_{2}=0,*D*=1.8652 and*r*=1.0842, the critical point giving the resonance is (*k*_{c},*R*_{1})=(2.9575, 1336.5). This implies that the : 1 resonance takes place for small*ϵ*_{1}and*ϵ*_{2}.↵We have checked the convergence of eigenfunction expansions, numerically. Changing the number of eigenfunctions from 10 to 25, we concluded that, with 20 eigenfunctions, all the coefficients converged within five decimal digits, at least.

- Received August 11, 2007.
- Accepted September 25, 2007.

- © 2007 The Royal Society