Extreme multiplicity in cylindrical Rayleigh-Bénard convection. II. Bifurcation diagram and symmetry classification

A large number of flows with distinctive patterns have been observed in experiments and simulations of Rayleigh-Bénard convection in a water-filled cylinder whose radius is twice the height. We have adapted a time-dependent pseudospectral code, first, to carry out Newton’s method and branch continuation and, second, to carry out the exponential power method and Arnoldi iteration to calculate leading eigenpairs and determine the stability of the steady states. The resulting bifurcation diagram represents a compromise between the tendency in the bulk toward parallel rolls and the requirement imposed by the boundary conditions that primary bifurcations be toward states whose azimuthal dependence is trigonometric. The diagram contains 17 branches of stable and unstable steady states. These can be classified geometrically as roll states containing two, three, and four rolls; axisymmetric patterns with one or two tori; threefold-symmetric patterns called Mercedes, Mitsubishi, marigold, and cloverleaf; trigonometric patterns called dipole and pizza; and less symmetric patterns called CO and asymmetric three rolls. The convective branches are connected to the conductive state and to each other by 16 primary and secondary pitchfork bifurcations and turning points. In order to better understand this complicated bifurcation diagram, we have partitioned it according to azimuthal symmetry. We have been able to determine the bifurcation-theoretic origin from the conductive state of all the branches observed at high Rayleigh number.

Figure 1

(Color online) Schematic diagram of existence ranges and transitions between convective patterns observed in time-dependent simulation for insulating sidewalls. Stars denote solutions obtained from a slight perturbation of the conductive state, at the Rayleigh numbers indicated. The initial condition was identical for all five simulations. Arrows indicate patterns obtained by starting from stable steady states, and abruptly either lowering or raising the Rayleigh number. For example, at $\text{Ra}=2000$, the perturbed conductive initial condition leads to a pizza state. Using the pizza state at $\text{Ra}=2000$ as an initial condition leads to a four-roll state at $\text{Ra}=5000$, to the three-roll state at $\text{Ra}=10000$, and to a two-roll state at $\text{Ra}\ge 15000$. Right: representative patterns illustrated via temperature field in the horizontal midplane, with light portions representing hot rising fluid and dark portions representing cold descending fluid.

Figure 2

(Color online) Bifurcation diagram containing 17 branches of steady states, in addition to the conductive branch (indicated by short-dashed horizontal line). Shown are pizza (solid green), four-roll (long-dashed turquoise), two-tori (solid red; 2), torus (long-dashed magenta; 2), marigold (solid blue), Mitsubishi (short-dashed purple), cloverleaf (long-dashed purple), and Mercedes (solid blue), three-roll (solid black), tiger (long-dashed brick), asymmetric three-roll (solid brick; 2), two-roll (solid blue; 2), CO (long-dashed red) branches. The notation (; 2) represents a pair of related branches, connected by a saddle-node bifurcation. Note that the bifurcation diagram gives no information concerning stability, i.e., whether a solution curve is depicted as solid or dashed does not indicate its stability. Turning points or pitchfork bifurcations are shown as dots.

Figure 3

(Color online) First branches bifurcating from conductive state. For this aspect ratio, $\Gamma =2$, and with insulating lateral walls, the first four critical wave number and Rayleigh number pairs are ($m=1$, $\text{Ra}=1828$; black curve), ($m=2$, $\text{Ra}=1849$; green curve), ($m=0$, $\text{Ra}=1861$; red curve), and ($m=3$, $\text{Ra}=1985$; blue curve). Below are representative states from each of the bifurcating branches.

Figure 4

(Color online) Partial bifurcation diagram including $m=2$ primary branch and connecting branches. Bifurcations are shown as dots. The primary pizza branch (solid green curve) bifurcates from the conductive state at $\text{Ra}=1849$ and terminates in a saddle-node bifurcation at $\text{Ra}\approx 19450$. The four-roll branch (dashed turquoise curve) bifurcates from the pizza branch at $\text{Ra}=2353$ and terminates at a saddle-node bifurcation at $\text{Ra}\approx 23130$.

Figure 5

(Color online) Schematic partial bifurcation diagram relating branches originating from the $m=2$ bifurcation. At $\text{Ra}=1849$ the pizza branch originates via a circle pitchfork bifurcation from the conductive state corresponding to an $m=2$ eigenvector. It terminates at a turning point at $\text{Ra}\lesssim 19450$ and is stable for $1879\le \text{Ra}\le 2353$. At $\text{Ra}=2353$, a secondary pitchfork bifurcation leads to a four-roll branch, which is stable for $\text{Ra}\lesssim 22660$ and ends at a turning point at $\text{Ra}\approx 23130$. For visual clarity, the Rayleigh numbers given for the representative states have been rounded to the nearest 10, 100, or 1000.

Figure 6

Four leading eigenvalues of (a) the pizza branch at low Ra and of (b) the four-roll branch at high Ra. Bifurcations (zero crossings) are indicated by dots. The zero eigenvalue (dotted curve) which exists throughout corresponds to the marginal stability to rotation of the pattern. (a) The bifurcating eigenvalue (short-dashed curve) decreases steeply from zero at onset, $\text{Ra}=1849$. The pizza branch is initially unstable since it inherits the unstable eigenvalue (solid curve) of the conducting branch due to the preceding $m=1$ bifurcation. This leading eigenvalue decreases with Ra, crossing zero at $\text{Ra}=1879$. Another eigenvalue (long-dashed curve) becomes positive at $\text{Ra}=2353$, accompanying the bifurcation to the four-roll branch. The stability interval of the pizza branch is $1879\le \text{Ra}\le 2353$. (b) The four-roll branch loses stability near $\text{Ra}=22660$.

Figure 7

(Color online) Partial bifurcation diagram including only axisymmetric states. The two-tori branches (solid red curve) emerge from pitchfork bifurcations from the conductive branch at $\text{Ra}=1861.5$ and 2328. They join and terminate at a turning point at $\text{Ra}=12711$. The one-torus branches (dashed magenta curve) emerge from a turning point at $\text{Ra}=3076$ and seem to be unconnected to the conductive state.

Figure 8

(Color online) Schematic partial bifurcation diagram showing axisymmetric branches. The two-tori branches are connected to the conductive branch via pitchfork bifurcations at $\text{Ra}=1862$ and 2328, and to each other via a turning point at $\text{Ra}=12711$. The upper two-tori branch is stable for $2300\le \text{Ra}\le 5438$. The one-torus branches are connected to each other via a turning point at $\text{Ra}=3076$. The upper one-torus branch is stable for $\text{Ra}\ge 4918$. Most states along the two-tori branches contain two concentric rolls, but states on the lower two-tori branch resemble those on the upper one-torus branch for $\text{Re}\le 3500$.

Figure 9

Five leading eigenvalues of upper axisymmetric branches, each labeled with its azimuthal wave number $m$. (a) The upper two-tori branch has two positive eigenvalues at onset at $\text{Ra}=1862$, which cross zero at $\text{Ra}=2116$ and 2300 and again at $\text{Ra}=5438$. It is stable for $2300\le \text{Ra}\le 5438$. (b) The upper one-torus branch has five positive eigenvalues at onset at $\text{Ra}=3076$. These cross zero at $\text{Ra}=3330$, 3438, 4408, 4582, and 4918, above which the branch is stable.

Figure 10

(Color online) Partial bifurcation diagram including only branches with threefold symmetry. The marigold branch (solid blue curve) arises at a pitchfork bifurcation from the conductive branch (short-dashed black curve) at $\text{Ra}=1985$. A secondary pitchfork bifurcation from the marigold branch at $\text{Ra}=4103$ gives rise to the Mitsubishi branch (short-dashed light purple curve). At a turning point at $\text{Ra}=18762$, it meets the cloverleaf branch (long-dashed dark purple curve). The Mercedes branch (solid blue curve) originates at another turning point at $\text{Ra}=4634$.

Figure 11

(Color online) Leading eigenvalue for each of the threefold-symmetric branches. From highest to lowest: marigold (solid blue curve), Mitsubishi (short-dashed light purple curve), cloverleaf (long-dashed dark purple curve), Mercedes (solid blue curve). Dots indicate bifurcations from conductive to marigold branch $\left(\text{Ra}=1985,\sigma \approx 1.46\right)$, to Mitsubishi branch $\left(\text{Ra}=4103,\sigma \approx 10\right)$, to cloverleaf branch, $\left(\text{Ra}=18762,\sigma \approx 27\right)$, to Mercedes branch $\left(\text{Ra}=4634,\sigma \approx 1.3\right)$, and final stabilization of Mercedes branch $\left(\text{Ra}=5503,\sigma =0\right)$.

Figure 12

(Color online) Schematic partial bifurcation diagram relating branches originating from the $m=3$ bifurcation. The four branches of steady states with threefold symmetry are the Mercedes, cloverleaf, Mitsubishi, and marigold branches. The marigold branch is created by an $m=3$ circle pitchfork bifurcation from the conductive branch at $\text{Ra}=1985$. It undergoes a pitchfork bifurcation at $\text{Ra}=4103$, leading to the Mitsubishi branch. A turning point at $\text{Ra}=18762$ leads to the cloverleaf branch, and another turning point at $\text{Ra}=4634$ to the Mercedes branch. Only the Mercedes branch is stable, for $\text{Ra}>5503$, as indicated by the thick line.

Figure 13

(Color online) Partial bifurcation diagram of branches created at the $m=1$ primary bifurcation. Two branches of dipole states are created at $\text{Ra}=1828$ and seem to be tangent to one another near the bifurcation. As Ra increases, the spatial forms along these branches evolve in divergent ways. One branch contains the three-roll states (solid black curve) and the other (dashed brick) contains the tiger states. An asymmetric three-roll branch (solid brick) is created at a subcritical pitchfork bifurcation at $\text{Ra}=22155$ and reverses direction at a saddle-node bifurcation at $\text{Ra}=21078$.

Figure 14

(Color online) Schematic bifurcation diagram of branches emanating from $m=1$ instability. Two branches bifurcate simultaneously at $\text{Ra}=1828$. Near the bifurcation, states along both branches have the form of a dipole. States along one branch evolve into a three-roll state, while those along the other branch evolve to a form called tiger. The three-roll state is stable for $3762\le \text{Ra}\le 20393$, and the tiger branch is never stable. The asymmetric three-roll branch is formed at $\text{Ra}=22125$, via a subcritical pitchfork bifurcation from the three-roll branch, reversing direction at a saddle-node bifurcation at $\text{Ra}=21078$.

Figure 15

(Color online) (a) Leading eigenvalues of the tiger (long-dashed brick) and three-roll (solid black curve) branches. The eigenvalues of the tiger branch grow rapidly as Ra increases, while the three-roll branch is weakly unstable until $\text{Ra}=3762$. The rapidly decreasing eigenvalue is that associated with the formation of these branches. (b) The tiger and three-roll branches have the same curvature near onset.

Figure 16

(Color online) Enlargement of bifurcation diagram in Fig. 13. The symmetric three-roll branch loses stability at $\text{Ra}=20393$. A subcritical bifurcation at $\text{Ra}=22125$ from the symmetric three-roll branch leads to the creation of the asymmetric three-roll branch, which is stabilized by a saddle-node bifurcation at $\text{Ra}=21078$.

Figure 17

(Color online) Partial bifurcation diagram containing the two-roll and CO branches. Two-roll branches (solid blue curve) arise via saddle-node bifurcation at $\text{Ra}=8677$. CO branch (dashed red curve) has been computed for $7167\le \text{Ra}\le 10348$. The apparent intersections (between the two-roll and CO branches and between the stable and unstable two-roll branches) are an artifact of the projection.

Figure 18

(Color online) Schematic partial bifurcation diagram showing the two-roll and CO branches. The two-roll branches originate at a turning point at $\text{Ra}=8677$ and have been computed for $\text{Ra}\le 30000$. The upper branch is stable for $\text{Ra}\le 28086$. The CO branch has been calculated in the range $7167\le \text{Ra}\le 10348$; it is stable for $\text{Ra}\le 10087$.

Figure 19

(Color online) Four-roll (dashed turquoise curve for $\text{Ra}\lesssim 23000$), asymmetric three-roll (solid brick for $\text{Ra}\gtrsim 21000$), and two-roll branches (solid blue curve) at high Ra.

Figure 20

Schematic bifurcation diagram. Arbitrary quantity of the vertical axis chosen to eliminate all but one intersection. Bold lines indicate stable portions of branches.