3.1.1 Bifurcation branch of skewed varicose states

When convection patterns in experiments or numerical simulations exhibit complex dynamics, we expect the existence of invariant states underlying the pattern dynamics. For the pattern dynamics emerging from the skewed varicose instability of straight convection rolls $R_{\unicode[STIX]{x1D706}}$ at $\unicode[STIX]{x1D6FE}=0^{\circ }$, however, we do not find invariant states at the control parameter values where the dynamics is observed. Direct numerical simulations in a minimal periodic domain can reproduce the transient dynamics of the skewed varicose pattern, but previous analysis of the temporal dynamics did not yield an underlying invariant state (Reference Reetz and SchneiderRS20).

An equilibrium state $SV$ resembling the observed skewed varicose pattern is identified below $\unicode[STIX]{x1D716}_{c}$ of the skewed varicose instability, as described in § 2.2. Numerical continuation of $SV$ reveals a subcritical $\unicode[STIX]{x1D716}$-backward pitchfork bifurcation from $R_{\unicode[STIX]{x1D706}2}$ at $\unicode[STIX]{x1D716}_{c}=1.019$. The bifurcation breaks the continuous translation symmetry $\unicode[STIX]{x1D70F}(a_{x},0)$ of straight convection rolls $R_{\unicode[STIX]{x1D706}2}$. Here we consider the rolls to be $x$-aligned and periodic with wavelength $\unicode[STIX]{x1D706}_{y}$. The bifurcating eigenmode shows a skewed three-dimensional flow structure. The bifurcating equilibrium $SV$ is $[4\unicode[STIX]{x1D706}_{x},4\unicode[STIX]{x1D706}_{y}]$-periodic and invariant under transformations of the symmetry group $S_{SV}=\langle \unicode[STIX]{x03C0}_{xyz},\unicode[STIX]{x1D70F}(0.25,0.25)\rangle$. From the bifurcation point, the $SV$ branch continues down in $\unicode[STIX]{x1D716}$, undergoes a sequence of folds, and terminates at $\unicode[STIX]{x1D716}=0.206$ in a bifurcation from straight convection rolls $R_{\unicode[STIX]{x1D706}3}$ with wavelength $\unicode[STIX]{x1D706}=2.8$ (panel $\unicode[STIX]{x1D6FE}=0^{\circ }$ in figure 3). Thus, the equilibrium state $SV$ connects two equilibrium states representing straight convection rolls at different wavelengths. The $SV$ state exists only below the critical threshold parameter $\unicode[STIX]{x1D716}_{c}$. The pure subcritical existence of $SV$ explains why no temporal transition to an underlying invariant state at $\unicode[STIX]{x1D716}>\unicode[STIX]{x1D716}_{c}$ has been found in Reference Reetz and SchneiderRS20.

Since the bifurcation branches are very cluttered at $\unicode[STIX]{x1D6FE}=0^{\circ }$ in figure 3, we reproduce the bifurcation diagram schematically. In figure 5 the bifurcation branches are plotted in terms of their approximate dominating pattern wavelength $\unicode[STIX]{x1D706}$ as a function of $\unicode[STIX]{x1D716}$. Along the $SV$ branch, convection rolls develop skewed relative orientations (figure 5b) until the rolls pinch-off and reconnect at an oblique orientation (figure 5c). At the bifurcation point, $R_{\unicode[STIX]{x1D706}3}$ is rotated by $74.6^{\circ }$ against the orientation of $R_{\unicode[STIX]{x1D706}2}$ (figure 5a,c). To link these two different roll orientations, the continuous deformations in the skewed varicose pattern skip two instances for potential reconnection to straight rolls with orientations in between. Each of the potential reconnection points corresponds to a pair of folds along the $SV$ branch. Here, three pairs of folds are observed but this number is specific to the chosen domain size. In between the first two folds at $0.94<\unicode[STIX]{x1D716}<0.95$, the $SV$ branch is bistable with $R_{\unicode[STIX]{x1D706}2}$ and $R_{\unicode[STIX]{x1D706}3}$ in a symmetry subspace of $S_{SV}$. The stability of all branches is indicated by the line style. Overall, the bifurcation diagram indicates coexistence of stable (or weakly unstable) $SV$ with stable $R_{\unicode[STIX]{x1D706}2}$ and $R_{\unicode[STIX]{x1D706}3}$ over a range of $\unicode[STIX]{x1D716}$. The coexistence of invariant states suggests a spatial coexistence of straight convection rolls and the skewed varicose patterns.

The convection pattern along the $SV$ branch at $\unicode[STIX]{x1D716}<\unicode[STIX]{x1D716}_{c}$ may be compared to the convection pattern observed transiently in time along a simulated transition at $\unicode[STIX]{x1D716}=1.05>\unicode[STIX]{x1D716}_{c}$ (dashed line in figure 5d and § 4.3 in Reference Reetz and SchneiderRS20). The midplane temperature contours of $SV$ along its subcritical bifurcation branch partly match the transient patterns along the supercritical temporal transition. We find matching patterns at initial instances in time when straight convection rolls are observed (figure 5c,g), and at an intermediate time when the transient pattern of skewed varicose pattern emerges (figure 5b,f). Thus, $SV$ indeed captures the observed transient pattern triggered by the skewed varicose instability, but the comparison is for different $\unicode[STIX]{x1D716}$. This observation raises the question of how the transient temporal dynamics observed above critical thresholds can be related to an equilibrium state existing only below critical thresholds.

Figure 5.

For $\unicode[STIX]{x1D6FE}=0^{\circ }$ (Rayleigh–Bénard), the subcritical bifurcation of the skewed varicose state connects two equilibrium states of straight convection rolls at different wavelength and orientation. Midplane temperature fields of equilibrium states (a–c) are chosen along the $SV$ branch, plotted here in a schematic bifurcation diagram in terms of an approximate dominating pattern wavelength $\unicode[STIX]{x1D706}$ over $\unicode[STIX]{x1D716}$ in (d). A simulated temporal transition from unstable $R_{\unicode[STIX]{x1D706}2}$ to stable $R_{\unicode[STIX]{x1D706}3}$ at supercritical $\unicode[STIX]{x1D716}>\unicode[STIX]{x1D716}_{c}$ (Reference Reetz and SchneiderRS20) is indicated by the dashed line. Snapshots from the temporal transition (g,f,e) show matching skewed varicose patterns between the equilibrium state (b) for $\unicode[STIX]{x1D716}<\unicode[STIX]{x1D716}_{c}$ and the transient state (f) at $\unicode[STIX]{x1D716}>\unicode[STIX]{x1D716}_{c}$. The orientation of $R_{\unicode[STIX]{x1D706}3}$ differs between the terminating bifurcation branch at $\unicode[STIX]{x1D716}=0.2$ (a) and the attracted temporal dynamics at $\unicode[STIX]{x1D716}=1.05$ (e).

3.2.1 Equivariant Hopf bifurcation from longitudinal rolls

Starting from the periodic orbit $SSW$, found at $[\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D716}]=[15^{\circ },1.5]$ in Reference Reetz and SchneiderRS20, we continue the branch down in $\unicode[STIX]{x1D6FE}$. The $SSW$ branch bifurcates from $LR$ at $[\unicode[STIX]{x1D6FE}_{c},\unicode[STIX]{x1D716}_{c}]=[8.5^{\circ },1.5]$. This bifurcation is a $\unicode[STIX]{x1D6FE}$-forward, supercritical Hopf bifurcation with a critical orbit period of $T_{\mathit{SSW}}=80.2$. The bifurcation corresponds to four complex eigenvalues crossing the imaginary axis at $\unicode[STIX]{x1D714}_{i}=\pm 0.078$ controlling the period $T_{\mathit{SSW}}=2\unicode[STIX]{x03C0}/|\unicode[STIX]{x1D714}_{i}|$. The Hopf bifurcation to $SSW$ accounts only for one pair of complex neutral eigenmodes. The other pair gives rise to a travelling wave state that we term subharmonic travelling wave ($STW$) with a critical phase speed of $c=L_{x}/T_{STW}=0.055$, where $T_{STW}=T_{\mathit{SSW}}$. Both invariant states, the $\unicode[STIX]{x03C0}_{y}$-symmetric $STW$ and the $\unicode[STIX]{x03C0}_{xyz}$-symmetric $SSW$ (shown in the left panels of figure 6a,b), each have a counter-propagating sibling state obtained via $\unicode[STIX]{x03C0}_{xz}$-transformation. Both invariant states capture a subharmonic oscillatory convection pattern invariant under $\unicode[STIX]{x1D70F}(0.5,0.5)$. An equilibrium pattern very close to $STW$ can be observed in spatially forced horizontal convection (Weiss, Seiden & Bodenschatz Reference Weiss, Seiden and Bodenschatz2012). The $STW$ state also resembles the subharmonic ‘sinucose’ state arising from an instability of longitudinal streaks in a pure shear flow (Waleffe Reference Waleffe1997).

At parameters where $SSW$ and $STW$ bifurcate locally from $LR$, they share the same bifurcation point (see panels for $10^{\circ }\leqslant \unicode[STIX]{x1D6FE}\leqslant 60^{\circ }$ in figure 3). This robust feature in the bifurcation diagram is a consequence of equivariant Hopf bifurcations. It is known that Hopf bifurcations that break the $O(2)$-symmetry of a flow must result in two branches originating from the bifurcation: a standing wave branch and a travelling wave branch (Knobloch Reference Knobloch1986). At most one of the two branches is stable. In the present case, the Hopf bifurcation breaks the $O(2)$-symmetry of the $x$-uniform state $LR$. The bifurcation at $[\unicode[STIX]{x1D6FE}_{c},\unicode[STIX]{x1D716}_{c}]=[8.5^{\circ },1.5]$ (inset panel in figure 6c) has both the branches bifurcating supercritically. The $SSW$ state is initially stable and $STW$ is unstable. This corresponds to one specific case of the six discussed in Knobloch (Reference Knobloch1986). However, here the bifurcation is secondary. While in Knobloch (Reference Knobloch1986) bifurcations from a non-patterned two-dimensional primary state with a spatial $O(2)$-symmetry are discussed, here, the bifurcating secondary equilibrium state $LR$ is a three-dimensional state that is symmetric under transformations of $O(2)\times Z_{2}$. The additional third dimension and the additional reflection symmetry do not affect the conditions necessary for equivariant Hopf bifurcations (Knobloch Reference Knobloch1986).

Continuations of $SSW$ and $STW$ in $\unicode[STIX]{x1D6FE}$ reveal their existence over a large range of inclination angles $\unicode[STIX]{x1D6FE}$, including $\unicode[STIX]{x1D6FE}>90^{\circ }$ where their parent state $LR$ does not exist anymore (figure 6c). Over this range in $\unicode[STIX]{x1D6FE}$, the orbit period of $SSW$ and the propagation time $T_{STW}=L_{x}/c$ of $STW$ with phase velocity $c$ and $L_{x}=2\unicode[STIX]{x1D706}_{x}$ follow approximately the mean laminar advection time $L_{x}/\bar{U}_{0}$ (figure 6d). While the state branch of $STW$ at $\unicode[STIX]{x1D716}=1.5$, shown as light grey line in figure 6(c), connects two Hopf bifurcations, one at small and one at large $\unicode[STIX]{x1D6FE}$, the $SSW$ branches at $\unicode[STIX]{x1D716}=1.5$, shown as black lines in figure 6(c), originating from these two bifurcations remain disconnected. The branch bifurcating forward at $\unicode[STIX]{x1D6FE}_{c}=8.5^{\circ }$ undergoes a fold at $\unicode[STIX]{x1D6FE}=15.2^{\circ }$ (figure 6c), explaining why no temporal state transition from $LR$ to $SSW$ was found at $\unicode[STIX]{x1D6FE}=17^{\circ }$ in Reference Reetz and SchneiderRS20. The fold destabilizes $SSW$ and connects to a state branch reaching to subcritical parameters where $LR$ is linearly stable. Similar folds occur under $\unicode[STIX]{x1D716}$ continuations at $\unicode[STIX]{x1D6FE}=10^{\circ }$ and $\unicode[STIX]{x1D6FE}=20^{\circ }$. Beyond these folds, $SSW$ state branches terminate and show ‘loose ends’ in the bifurcation diagrams. These terminations correspond to global bifurcations which we explain in the next subsection.

3.2.2 Global bifurcation to subharmonic standing waves

The global bifurcation at $\unicode[STIX]{x1D6FE}=10^{\circ }$ occurs at $Ra_{c}=2230.25$ ($\unicode[STIX]{x1D716}_{c}=0.286$), where the pre-periodic orbit $SSW$, satisfying (2.7) with $\unicode[STIX]{x1D70E}_{\mathit{SSW}}=\unicode[STIX]{x03C0}_{y}\unicode[STIX]{x1D70F}(0.25,0.25)$ and a pre-period of $T^{\prime }=T/4$, collides with a heteroclinic cycle between two symmetry related saddle states $TR$ and $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ (figure 7). At the global bifurcation point, the spectrum of eigenvalues of $TR$ is computed in a symmetry subspace $\unicode[STIX]{x1D6F4}_{0}$ given by the $[2\unicode[STIX]{x1D706}_{x},2\unicode[STIX]{x1D706}_{y}]$-periodic domain with imposed symmetries of $SSW$, namely $\unicode[STIX]{x1D70F}(0.5,0.5)$ and $\unicode[STIX]{x03C0}_{xyz}$. The five leading eigenvalues are real and have the specific numerical values $[\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3},\unicode[STIX]{x1D714}_{4},\unicode[STIX]{x1D714}_{5}]$$=$$[0.048,0.045,-0.090,-0.120,-0.138]$. The midplane temperature contours of the associated eigenmodes $[\boldsymbol{e}_{1}^{u},\boldsymbol{e}_{2}^{u},\boldsymbol{e}_{3}^{s}\boldsymbol{e}_{4}^{s},\boldsymbol{e}_{5}^{s}]$ are given in figure 7(h). Eigenvalues and eigenmodes of $TR$ do not change significantly when $Ra$ crosses $Ra_{c}$. In contrast to the heteroclinic cycle discussed in Reference Reetz and SchneiderRS20, where each of the two symmetry related instances of equilibrium state $OWR$ (further discussed in § 3.3.2) has a single unstable eigenmode, the present cycle connects symmetry related instances of $TR$ with two unstable eigenmodes each. Perturbations of $TR$ with the eigenmode $\boldsymbol{e}_{1}^{u}$ trigger a state transition $TR\rightarrow \unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$, while perturbations with $\boldsymbol{e}_{2}^{u}$ lead to $LR$ which is dynamically stable in $\unicode[STIX]{x1D6F4}_{0}$ at these control parameter values. The symmetry relation between $TR$ and $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ guarantees that $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ has the same eigenvalues as $TR$ and symmetry related eigenmodes $\unicode[STIX]{x1D70E}_{\mathit{SSW}}[\boldsymbol{e}_{1}^{u},\boldsymbol{e}_{2}^{u},\boldsymbol{e}_{3}^{s},\boldsymbol{e}_{4}^{s},\boldsymbol{e}_{5}^{s}]$, allowing for the returning transition $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR\rightarrow TR$ to close the heteroclinic cycle.

Figure 7.

The pre-periodic orbit $SSW$ approaches a global bifurcation at $\unicode[STIX]{x1D6FE}=10^{\circ }$ and $Ra_{c}=2230.25$ ($\unicode[STIX]{x1D716}_{c}=0.286$), where it collides with a robust heteroclinic cycle between $TR$ and the symmetry related equilibrium $\unicode[STIX]{x1D70E}_{SSW}TR$ with $\unicode[STIX]{x1D70E}_{SSW}=\unicode[STIX]{x03C0}_{y}\unicode[STIX]{x1D70F}(0.25,0.25)$. (a–e) Sequence of midplane temperature contours along the dynamical connection $TR\rightarrow \unicode[STIX]{x1D70E}_{SSW}TR$ at $Ra=2230$. (f) State space projection of the lower (LB) and the upper branch (UB) of $SSW$ (cf. figure 8a). Inset enlarges the orbit trajectories (black dots) near $TR$ and the heteroclinic cycle (grey lines). (g) $L_{2}$-distance of the $SSW_{UB}$ orbit trajectory from $TR$ at $Ra=2230.24$ very close to $Ra_{c}$. The dynamics of $SSW_{UB}$ is exponential for most of the pre-period and governed by two eigenvalues of $TR$. (h) Midplane temperature contours of two unstable and three stable eigenmodes $\boldsymbol{e}_{i}^{u/s}$ of $TR$ associated to the five leading eigenvalues $[\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}_{2},\unicode[STIX]{x1D714}_{3},\unicode[STIX]{x1D714}_{4},\unicode[STIX]{x1D714}_{5}]=[0.048,0.045,-0.090,-0.120,-0.138]$ of $TR$.

Direct numerical simulations indicate that states close to the heteroclinic cycle are eventually attracted to $LR$. To show that this heteroclinic cycle is dynamically unstable but structurally stable, we identify two symmetry subspaces of $\unicode[STIX]{x1D6F4}_{0}$ in which either $TR\rightarrow \unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ or $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR\rightarrow TR$ exists as a heteroclinic connection between an equilibrium with a single unstable eigenmode and a dynamically stable equilibrium. By doing so, the heteroclinic cycle is shown to satisfy all conditions of a structurally stable, or robust, heteroclinic cycle between two symmetry related equilibrium states (Krupa Reference Krupa1997), also discussed in Reference Reetz and SchneiderRS20. Subspace $\unicode[STIX]{x1D6F4}_{1}$ is given by imposing the symmetries in the group $\langle \unicode[STIX]{x03C0}_{y},\unicode[STIX]{x03C0}_{xz},\unicode[STIX]{x1D70F}(0.5,0.5)\rangle$ and contains the connection $TR\rightarrow \unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$. Of the five initially considered eigenmodes in $\unicode[STIX]{x1D6F4}_{0}$, $TR$ in $\unicode[STIX]{x1D6F4}_{1}$ has still $[\boldsymbol{e}_{1}^{u},\boldsymbol{e}_{3}^{s},\boldsymbol{e}_{5}^{s}]$ and $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ in $\unicode[STIX]{x1D6F4}_{1}$ has still $\unicode[STIX]{x1D70E}_{\mathit{SSW}}[\boldsymbol{e}_{4}^{s},\boldsymbol{e}_{5}^{s}]$. Subspace $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70F}}$ is given by imposing the symmetries in the group $\langle \unicode[STIX]{x03C0}_{y}\unicode[STIX]{x1D70F}(0.5,0),\unicode[STIX]{x03C0}_{xz}\unicode[STIX]{x1D70F}(0.5,0),\unicode[STIX]{x1D70F}(0.5,0.5)\rangle$ and contains the connection $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR\rightarrow TR$. Of the five initially considered eigenmodes in $\unicode[STIX]{x1D6F4}_{0}$, $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ in $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70F}}$ has still $\unicode[STIX]{x1D70E}_{\mathit{SSW}}[\boldsymbol{e}_{1}^{u},\boldsymbol{e}_{3}^{s},\boldsymbol{e}_{5}^{s}]$ and $TR$ in $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70F}}$ has still $[\boldsymbol{e}_{4}^{s},\boldsymbol{e}_{5}^{s}]$. Using the classification of eigenvalues and associated stability theorem in Krupa & Melbourne (Reference Krupa and Melbourne1995), we identify $\unicode[STIX]{x1D714}_{1}$ as expanding, $\unicode[STIX]{x1D714}_{2}$ as transverse, $\unicode[STIX]{x1D714}_{4}$ as contracting and $\unicode[STIX]{x1D714}_{5}$ as a radial eigenvalue. Eigenvalue $\unicode[STIX]{x1D714}_{3}$ exists in the same subspace as the expanding eigenvalue $\unicode[STIX]{x1D714}_{1}$ and, therefore, does not affect the dynamical stability. Since $\min (-\unicode[STIX]{x1D714}_{4},\unicode[STIX]{x1D714}_{1}-\unicode[STIX]{x1D714}_{2})\ngtr \unicode[STIX]{x1D714}_{1}$, the heteroclinic cycle is not asymptotically stable (Krupa & Melbourne Reference Krupa and Melbourne1995, Theorem 2.7).

Before $SSW$ disappears in the global bifurcation at $Ra_{c}$, the solution branch undergoes a fold at $Ra<Ra_{c}$ (figure 8a). The existence of such a fold near a global bifurcation follows from the dynamical stability of the bifurcating periodic orbit relative to the dynamical stability of the heteroclinic cycle. To analyse the stability, we consider the linearized dynamics around the heteroclinic cycle $TR\rightarrow \unicode[STIX]{x1D70E}_{\mathit{SSW}}TR\rightarrow TR$ at $Ra_{c}$ and obtain the Poincaré map (see Bergeon & Knobloch Reference Bergeon and Knobloch2002, for a derivation)

(3.7a,b)$$\begin{eqnarray}\unicode[STIX]{x1D701}_{i+1}=c\unicode[STIX]{x1D701}_{i}^{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D707},\quad \unicode[STIX]{x1D70C}=-\frac{\unicode[STIX]{x1D714}_{4}}{\unicode[STIX]{x1D714}_{1}}=2.51,\end{eqnarray}$$

with constant $c>0$ and control parameter $\unicode[STIX]{x1D707}\propto Ra_{c}-Ra$. Variable $\unicode[STIX]{x1D701}_{i}\ll 1$ describes a local coordinate in a Poincaré section located at a distance $\unicode[STIX]{x1D700}\ll 1$ from $TR$ and defining a small perturbation around the state vector of $TR$ as $\boldsymbol{x}^{\prime }=\boldsymbol{x}_{TR}+\unicode[STIX]{x1D701}_{i}\boldsymbol{e}_{1}^{u}+\unicode[STIX]{x1D700}\boldsymbol{e}_{4}^{s}$. The heteroclinic cycle corresponds to $\unicode[STIX]{x1D701}_{i}=0$ and is reached at $\unicode[STIX]{x1D707}=0$. The bifurcating periodic orbit is a fixed point $\bar{\unicode[STIX]{x1D701}}$ of the map such that $\bar{\unicode[STIX]{x1D701}}=c\bar{\unicode[STIX]{x1D701}}^{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D707}$. Since $\unicode[STIX]{x1D70C}>1$, a nearby fixed point $\bar{\unicode[STIX]{x1D701}}$ exists only for $\unicode[STIX]{x1D707}>0$ and $Ra<Ra_{c}$, respectively. The graph in figure 8(b) illustrates the map (3.7) and shows that the fixed point $\bar{\unicode[STIX]{x1D701}}$ is dynamically stable. The stability of $\bar{\unicode[STIX]{x1D701}}$ assumes that no additional transverse eigendirections are unstable. However, the symmetry subspace that contains $SSW$ also contains $\boldsymbol{e}_{2}^{u}$, which must be taken into account. Thus, the above analysis predicts that $SSW$ bifurcates with a single unstable eigendirection from the heteroclinic cyle, namely $\boldsymbol{e}_{2}^{u}$. Since $SSW$ for $Ra>Ra_{c}$ has two unstable eigenmodes, the periodic orbit must undergo a fold prior to the global bifurcation to stabilize the extra unstable eigendirection. Such a fold also exists at $\unicode[STIX]{x1D6FE}=20^{\circ }$ (figure 3), but further away from the global bifurcation. Note that unlike the global bifurcation discussed in Bergeon & Knobloch (Reference Bergeon and Knobloch2002), the present bifurcation involves first, a heteroclinic cycle between two symmetry related equilibrium states and second, a dynamically unstable periodic orbit such that the fold may have a stabilising effect.

Figure 8.

Before $SSW$ disappears in a global heteroclinic bifurcation at $\unicode[STIX]{x1D6FE}=10^{\circ }$ and $Ra_{c}=2230.25$ ($\unicode[STIX]{x1D716}_{c}=0.286$), the branch undergoes a saddle node bifurcation. (a) Enlarged bifurcation diagram showing the branch of $SSW$ indicating the maximum $\Vert \unicode[STIX]{x1D703}\Vert _{2}$ over one cycle (cf. panel $\unicode[STIX]{x1D6FE}=10^{\circ }$ in figure 3). (b) Graphical representation of the linearized Poincaré map (3.7) for $\unicode[STIX]{x1D707}=0.1$ around the heteroclinic cycle between symmetry related instances of $TR$. The fixed point $\bar{\unicode[STIX]{x1D701}}$ is stable as the slope of the map at $\bar{\unicode[STIX]{x1D701}}$ is less than one (dashed line). (c,d) Towards the global bifurcation, the orbit period of $SSW$ approaches an infinite period according to a logarithmic scaling law shown as a function of $Ra$ in (c) and for rescaled $Ra$ in (d). This law follows from the linearized map (3.7), as described in the text. The constant $c_{T}=560$ is determined by fitting the data close to $Ra_{c}$.

The period of $SSW$ must increase towards an infinite time period as the periodic orbit approaches the heteroclinic cycle. The map (3.7) suggests an asymptotic scaling law for $T$ as a function of $Ra$ close to the global bifurcation. Since $\unicode[STIX]{x1D701}\ll 1$ and $\unicode[STIX]{x1D70C}>1$, the periodic orbit is given by the approximation $\bar{\unicode[STIX]{x1D701}}\approx \unicode[STIX]{x1D707}$. Over a full period of $SSW$, the orbit trajectory visits both $TR$ and $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ twice. The time the orbit trajectory spends in the $\unicode[STIX]{x1D700}$-neighbourhood of $TR$ or $\unicode[STIX]{x1D70E}_{\mathit{SSW}}TR$ dominates the entire orbit period $T$ (figure 7g) such that $T$ satisfies the approximation $\bar{\unicode[STIX]{x1D701}}\approx \unicode[STIX]{x1D700}\exp (-\unicode[STIX]{x1D714}_{1}T/4)$. Hence, the period of $SSW$ is expected to increase as $T\approx -4/\unicode[STIX]{x1D714}_{1}\ln (Ra_{c}-Ra)+c_{T}$ with constant $c_{T}=4\ln (\unicode[STIX]{x1D700})/\unicode[STIX]{x1D714}_{1}$ so that asymptotically the period scales as $T\sim -4/\unicode[STIX]{x1D714}_{1}\ln (Ra_{c}-Ra)$. Using a multi-shooting method, $SSW$ is continued close to the global bifurcation. The increasing orbit period confirms the predicted scaling law (figure 8c,d).

3.3.1 Pitchfork bifurcations from longitudinal rolls

Equilibrium states $WR$ emerge either in pitchfork bifurcations from $LR$ at inclinations $20^{\circ }\leqslant \unicode[STIX]{x1D6FE}\leqslant 80^{\circ }$ or in saddle-node bifurcations in the absence of $LR$ at $90^{\circ }\leqslant \unicode[STIX]{x1D6FE}\leqslant 100^{\circ }$. The fact that $WR$ can exist without bifurcating from $LR$ is known from thermal Couette flow (Clever & Busse Reference Clever and Busse1992). Almost all computed pitchfork bifurcations from $LR$ to $WR$ are either $\unicode[STIX]{x1D716}$-forward or $\unicode[STIX]{x1D6FE}$-backward. This observation holds even when the $WR$ branches develop additional folds, like in the bifurcation diagrams at $\unicode[STIX]{x1D6FE}=80^{\circ }$ or at $\unicode[STIX]{x1D716}=1.5$. The only $\unicode[STIX]{x1D6FE}$-forward bifurcation of $WR$ from $LR$ is observed at $\unicode[STIX]{x1D716}=0.1$ (figure 4, $\unicode[STIX]{x1D716}=0.1$).

3.3.2 Additional bifurcations from wavy rolls

In most cases, $WR$ branches continue to large $\unicode[STIX]{x1D716}$. This does not imply the absence of connections to other invariant states. When $WR$ lose stability, additional bifurcations occur. We demonstrate the increasing number of invariant states and their patterns by following the higher-order instabilities of $WR$ at $\unicode[STIX]{x1D6FE}=40^{\circ }$. Arnoldi iteration for $WR$ indicates the stability of the $WR$ branch up to $\unicode[STIX]{x1D716}=0.256$ in a $[2\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{y}]$-periodic domain (figure 9g). At this point, a subcritical pitchfork bifurcation breaks the $\unicode[STIX]{x03C0}_{xz}$- and $\unicode[STIX]{x03C0}_{y}$-symmetry and gives rise to an equilibrium state showing disconnected wavy rolls and named $DWR$. The $DWR$ state is invariant under $\unicode[STIX]{x03C0}_{xyz}$. Following the $DWR$ branch from the pitchfork bifurcation, it undergoes a saddle-node bifurcation at $\unicode[STIX]{x1D716}=0.254$, becomes bistable with $WR$ and connects to an equilibrium state of oblique rolls that continues as a stable branch to larger values of $\unicode[STIX]{x1D716}$. The $DWR$ state represents stationary roll defects that along its branch breaks the topology of rolls in a double periodic domain (figure 9b), and connects convection rolls at different orientations.

Figure 9.

Bifurcation sequence from wavy rolls to oblique wavy rolls (a–d). Bifurcation diagram (e) shows how seven equilibrium states are connected over $\unicode[STIX]{x1D716}$ and indicates stable (solid) and unstable (dotted) parts of the branches. Dynamical stability is computed by Arnoldi iteration in a $(2\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{y})$-periodic domain. Panel (g) enlarges the parameter region where $DWR$ connects $WR$ with $OR$. Panel (f) enlarges the parameter region where $OWR_{1}$ with pattern wavenumber $m=1$ connects $OWR$ with pattern wavenumber $m=2$ with $OR$. Here ‘RS20’ labels the control parameter value at which the temporal evolution has been studied in Reference Reetz and SchneiderRS20.

The $OR$ state is a secondary state bifurcating from the laminar flow $B$. The oblique orientation at angle $\tan ^{-1}(\unicode[STIX]{x1D706}_{y}/2\unicode[STIX]{x1D706}_{x})$ against the laminar flow direction coincides with the diagonal of the $[2\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{y}]$-periodic domain (figure 9c). The $OR$ state is invariant under transformations of $S_{OR}=\langle \unicode[STIX]{x03C0}_{xyz},\unicode[STIX]{x1D70F}(a_{x},\pm a_{x})\rangle$, with $a_{x}\in [0,1)$ corresponding to an$O(2)$-symmetric state. The sign in front of the continuous shift factor $a_{x}$ differs between left $(+)$ and right $(-)$ oblique rolls, with $OR^{l}=\unicode[STIX]{x03C0}_{y}OR^{r}$. Here, the superscripts ‘$l$’ and ‘$r$’ denote left and right, respectively. When $OR$ becomes unstable, an $\unicode[STIX]{x1D716}$-forward pitchfork bifurcation at $\unicode[STIX]{x1D716}=0.456$ gives rise to stable oblique wavy rolls, see figure 9(d). As $OR$, $OWR$ can have a left or right orientation. The symmetry group of $OWR$ is $S_{OWR}=\langle \unicode[STIX]{x03C0}_{xyz},\unicode[STIX]{x1D70F}(0.5,\pm 0.5)\rangle$. Equilibrium $OWR$ with a wavy pattern of wavenumber $m=2$ along the domain diagonal loses stability at $\unicode[STIX]{x1D716}=0.463$ to an equilibrium $OWR_{1}$ with a pattern wavenumber of $m=1$ and broken $\unicode[STIX]{x1D70F}(0.5,\pm 0.5)$-symmetry. The branch of $OWR_{1}$ undergoes a saddle-node bifurcation at $\unicode[STIX]{x1D716}=0.477$ and terminates on $OR$ at $\unicode[STIX]{x1D716}=0.476$ (figure 9f). The small $\unicode[STIX]{x1D716}$-range with $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=0.476-0.456=0.02$ between the two symmetry-breaking bifurcations of $OWR_{1}$ with $m=1$ and $OWR$ with $m=2$ from $OR$ suggests a nearby codimension-2 point with spatial 1 : 2 resonance. When stable $OWR_{1}$ disappear in the saddle-node bifurcation, the temporal dynamics becomes attracted to a robust heteroclinic cycle between unstable instances of $OWR$ that is discussed in Reference Reetz and SchneiderRS20. The branch of $OWR$ continues as an unstable branch until it terminates at $\unicode[STIX]{x1D716}=0.8$ on ribbons ($RB$), an unstable equilibrium state bifurcating together with $OR$ from $B$ in an equivariant pitchfork bifurcation (figure 9e). The detailed properties of $RB$ are discussed in the following section, but we already note that $RB$ shows disconnected rolls or plumes, similar to $DWR$. Thus, we find two instances of bifurcation sequences that may be described as ‘straight rolls bifurcate to wavy rolls, wavy rolls bifurcate to disconnected rolls’. In one instance the sequence happens for longitudinal orientation and in the other instance for oblique orientation. The fact that all of the above states coexist with the WR branch at equal control parameters explains that all patterns represented by the invariant states can spatially coexist with wavy rolls in large domains.

3.4.1 Bifurcations to states for knots and ribbons

The $KN$ state at $\unicode[STIX]{x1D6FE}=80^{\circ }$ bifurcates $\unicode[STIX]{x1D716}$-forward from $TR$. At smaller $\Vert \unicode[STIX]{x1D703}\Vert _{2}$ than $TR$, $KN$ continues without folds and terminates in $\unicode[STIX]{x1D716}$-backward bifurcations from $LR$ (figure 3, $\unicode[STIX]{x1D6FE}=80^{\circ }$). This bifurcation sequence requires $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{c2}$ and was previously analysed using two-mode interactions (Fujimura & Kelly Reference Fujimura and Kelly1993). At $\unicode[STIX]{x1D6FE}=90^{\circ }$, $LR$ does not exist at finite $\unicode[STIX]{x1D716}$ and the $KN$ branch terminates in a bifurcation from an equilibrium state we term subharmonic lambda plumes and briefly discuss in appendix A.

The $RB$ state is an equilibrium state found via continuing the state branches of $OWR$ and $WR$ that terminate in $\unicode[STIX]{x1D716}$-bifurcations from $RB$ at $\unicode[STIX]{x1D6FE}=40^{\circ }$ and $\unicode[STIX]{x1D6FE}=50^{\circ }$, respectively (figure 3). The $RB$ state is invariant under transformations of $S_{RB}=\langle \unicode[STIX]{x03C0}_{y},\unicode[STIX]{x03C0}_{xz},\unicode[STIX]{x1D70F}(0.5,0.5)\rangle$. Neither experiments nor simulations of ILC observe the pattern of $RB$ as a dynamically stable pattern at the considered parameters. However, we refer to experimental observations of ‘ribbons’ in Taylor–Couette flow (Tagg et al. Reference Tagg, Edwards, Swinney and Marcus1989). Ribbons in Taylor–Couette flow are analogous to ribbons in ILC. In Taylor–Couette flow, they bifurcate together with oblique spirals (Chossat & Iooss Reference Chossat and Iooss1994) and are connected via oblique wavy cross-spirals (Pinter, Lücke & Hoffmann Reference Pinter, Lücke and Hoffmann2006), two states that are comparable to $OR$ and $OWR$ in ILC. As in the Taylor–Couette flow, $OR$ and $RB$ in ILC bifurcate robustly together in equivariant bifurcations (Knobloch Reference Knobloch1986) and are connected via $OWR$. However, $OR$ and $RB$ are stationary states and their bifurcation is an equivariant pitchfork bifurcation, unlike equivariant Hopf bifurcations such as those found in Taylor–Couette flow, and those discussed in § 3.2.

3.4.2 Decomposition in terms of straight convection rolls

As a consequence of the stationary equivariant bifurcation, the linear relation $\boldsymbol{x}_{RB}=\unicode[STIX]{x1D6FC}\boldsymbol{x}_{OR}^{l}+\unicode[STIX]{x1D6FD}\boldsymbol{x}_{OR}^{r}$ with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$ holds at the bifurcation points, where $\boldsymbol{x}_{RB}$ indicates the state space vector of $RB$, and $\boldsymbol{x}_{OR}^{l}$ and $\boldsymbol{x}_{OR}^{r}$ are the state space vectors of $OR^{l}$ and $OR^{r}$, respectively. This linear decomposition is valid for all parameters, where $RB$ and $OR$ bifurcate from laminar flow $B$. Since $RB$ emerges as a linear superposition of two differently oriented straight convection rolls, we call $RB$ a ‘bimodal state’. The term ‘bimodal’ has been used previously to describe knot patterns of straight convection rolls at orthogonal orientations in experiments of Rayleigh–Bénard convection (Busse & Whitehead Reference Busse and Whitehead1974) and in experiments of ILC for $\unicode[STIX]{x1D6FE}>\unicode[STIX]{x1D6FE}_{c2}$ (Daniels et al. Reference Daniels, Plapp and Bodenschatz2000). In line with previously used terminology, we describe $KN$ and $RB$ both as bimodal states. However, there are fundamental differences between $KN$ and $RB$ bimodal states that are illustrated by the subsequently discussed decomposition analysis.

We consider a bimodal equilibrium state vector $\boldsymbol{b}(\unicode[STIX]{x1D716})$ depending on continuation parameter $\unicode[STIX]{x1D716}$ as decomposition

(3.8)$$\begin{eqnarray}\boldsymbol{b}(\unicode[STIX]{x1D716})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D716})\boldsymbol{m}_{1}(\unicode[STIX]{x1D716})+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D716})\boldsymbol{m}_{2}(\unicode[STIX]{x1D716})+\boldsymbol{d}(\unicode[STIX]{x1D716}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \mathbb{R}$ and $\boldsymbol{m}_{1},\boldsymbol{m}_{2}$ are state vectors of two differently oriented straight convection rolls, and $\boldsymbol{d}$ is the difference vector that is necessary to create the composite state $\boldsymbol{b}$. We simplify the notation by suppressing the dependence of the decomposition on $\unicode[STIX]{x1D716}$. We seek the optimal coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ such that $\Vert \boldsymbol{d}\Vert _{2}$ is minimal. The optimality condition $0=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}\Vert \boldsymbol{d}\Vert _{2}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FD}}\Vert \boldsymbol{d}\Vert _{2}$ implies that $\langle \boldsymbol{m}_{1},\boldsymbol{d}\rangle =0$ and $\langle \boldsymbol{m}_{2},\boldsymbol{d}\rangle =0$, where the inner product $\langle ,\rangle$ is induced by the full norm (2.11). As a consequence, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ may be found via inner products of (3.8) with $\boldsymbol{m}_{\mathbf{1}}$ and $\boldsymbol{m}_{\mathbf{2}}$, respectively,

(3.9a,b)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{\langle \boldsymbol{m}_{1},\boldsymbol{b}\rangle }{\langle \boldsymbol{m}_{1},\boldsymbol{m}_{1}\rangle }-\unicode[STIX]{x1D6FD}\frac{\langle \boldsymbol{m}_{1},\boldsymbol{m}_{2}\rangle }{\langle \boldsymbol{m}_{1},\boldsymbol{m}_{1}\rangle },\quad \unicode[STIX]{x1D6FD}=\frac{\langle \boldsymbol{m}_{2},\boldsymbol{b}\rangle }{\langle \boldsymbol{m}_{2},\boldsymbol{m}_{2}\rangle }-\unicode[STIX]{x1D6FC}\frac{\langle \boldsymbol{m}_{2},\boldsymbol{m}_{1}\rangle }{\langle \boldsymbol{m}_{2},\boldsymbol{m}_{2}\rangle }.\end{eqnarray}$$

These coupled equations determine the optimal coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$. The corresponding minimal $\boldsymbol{d}$ measures nonlinear and non-bimodal effects.

Figure 10.

Decomposition of bimodal ribbons into left and right oriented oblique rolls ($OR^{l,r}$). The bifurcation diagram with details of the decomposition (left) is shown together with visualizations of the temperature and velocity fields on the right (red labels in the diagram indicate panels on the right). The $RB$ state bifurcates together with $OR^{l,r}$ in equivariant pitchfork bifurcations from the laminar base flow. At $\unicode[STIX]{x1D6FE}=40^{\circ }$, $RB$ emerges with one additional unstable eigendirection, and $OR^{l,r}$ become dynamically stable in a $[2\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{y}]$-periodic domain at $\unicode[STIX]{x1D716}=0.282$ (solid line in top left panel). The optimal decomposition $\boldsymbol{x}_{RB}(\unicode[STIX]{x1D716})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D716})\boldsymbol{x}_{OR}^{l}(\unicode[STIX]{x1D716})+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D716})\boldsymbol{x}_{OR}^{r}(\unicode[STIX]{x1D716})+\boldsymbol{d}(\unicode[STIX]{x1D716})$ implies linearly decreasing equal coefficients $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$ along the branches (middle panel). The difference vector $\boldsymbol{d}$ grows linearly in $\Vert \unicode[STIX]{x1D703}\Vert _{2}$ from zero at the bifurcation point (bottom panel). Thus, $RB$ can be viewed as a bimodal state combining two equally contributing oblique rolls.

The optimal bimodal decomposition (3.8) with (3.9) is calculated for $\boldsymbol{b}=\boldsymbol{x}_{RB}$ and $\boldsymbol{m}_{1,2}=\boldsymbol{x}_{OR}^{l,r}$ along the $\unicode[STIX]{x1D716}$-bifurcation branches at $\unicode[STIX]{x1D6FE}=40^{\circ }$. The coefficients are found to be equal at all $\unicode[STIX]{x1D716}$, and to decrease linearly from $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}=0.3291$ at the bifurcation point (figure 10). The difference vector $\boldsymbol{d}$ increases linearly in $\Vert \unicode[STIX]{x1D703}\Vert _{2}$ and mostly accounts for corrections to the flow at the streamwise interfaces between hot and cold plumes (figure 10d).

Figure 11.

Decomposition of knots into longitudinal and transverse oriented rolls, $LR$ and $TR$, respectively. The bifurcation diagram with details of the decomposition (left) is shown together with visualizations of the temperature and velocity fields on the right (red labels in the diagram indicate panels on the right). The $KN$ state connects $TR$ and $LR$ at $\unicode[STIX]{x1D6FE}=80^{\circ }$ (top). Dynamical stability in a $[\unicode[STIX]{x1D706}_{x},\unicode[STIX]{x1D706}_{y}]$-periodic domain is indicated by the solid lines. Here ‘RS20’ labels the control parameter value where the temporal evolution has been studied in Reference Reetz and SchneiderRS20. Between the bifurcation points, the optimal decomposition $KN(\unicode[STIX]{x1D716})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D716})TR(\unicode[STIX]{x1D716}=0.024)+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D716})LR(\unicode[STIX]{x1D716}=0.22)+\boldsymbol{d}(\unicode[STIX]{x1D716})$ results in decreasing $\unicode[STIX]{x1D6FC}$, increasing $\unicode[STIX]{x1D6FD}$ (middle) and parabolically varying $\boldsymbol{d}$ (bottom). Thus, $KN$ can be viewed as a bimodal state combining transverse and longitudinal rolls with monotonically changing relative contributions.

The optimal bimodal decomposition (3.8) with (3.9) is calculated again for $\boldsymbol{b}=\boldsymbol{x}_{KN}$ along the $KN$ branch at $\unicode[STIX]{x1D6FE}=80^{\circ }$. Since $LR$ does not coexist with most of the $KN$ branch (figure 11), the state vectors $\boldsymbol{m}_{1}=\boldsymbol{x}_{TR}$ and $\boldsymbol{m}_{2}=\boldsymbol{x}_{LR}$ in the decomposition are not considered as $\unicode[STIX]{x1D716}$-dependent. We choose the decomposition $\boldsymbol{x}_{KN}(\unicode[STIX]{x1D716})=\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D716})\boldsymbol{x}_{TR}(\unicode[STIX]{x1D716}=0.024)+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D716})\boldsymbol{x}_{LR}(\unicode[STIX]{x1D716}=0.22)+\boldsymbol{d}(\unicode[STIX]{x1D716})$. Here, $\unicode[STIX]{x1D716}$ parametrizes linear interpolation between two bifurcation points. The resulting optimal coefficients $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ in general differ. While the contribution of the longitudinal rolls monotonically increases, the contribution of the transverse rolls decreases (figure 11). A decomposition with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$ is found at $\unicode[STIX]{x1D716}=0.095$, approximately half-way between the bifurcation points and close to the maximum of $\Vert \boldsymbol{d}\Vert _{2}$. Note that $\boldsymbol{d}$ at the maximum amplitude resembles a ribbon pattern (figure 11d). Towards the bifurcation points, $\boldsymbol{d}$ decreases parabolically to zero. Since $\boldsymbol{d}$ combines nonlinear and non-bimodal effects, as well as effects due to interpolation between $TR$ and $LR$ at fixed values of $\unicode[STIX]{x1D716}$, the dominant source for the large values of $\unicode[STIX]{x2202}\Vert \boldsymbol{d}\Vert _{2}/\unicode[STIX]{x2202}\unicode[STIX]{x1D716}$ at the bifurcations is unclear. Analysis in the weakly nonlinear regime near the knot instability indicates a three-mode interaction (Fujimura & Kelly Reference Fujimura and Kelly1993; Subramanian et al. Reference Subramanian, Brausch, Daniels, Bodenschatz, Schneider and Pesch2016), suggesting that beside longitudinal and transverse states a third resonant oblique state is involved. Here, the importance of an oblique mode is evidenced by the significant contribution of $\boldsymbol{d}$ (figure 11).

The states $KN$ and $RB$ differ in their bifurcation structure. The $KN$ state is a connecting state between $TR$ and $LR$, while $RB$ bifurcates together with $OR^{l,r}$ in an equivariant pitchfork bifurcation. As a consequence, their bimodal decomposition into two composing straight convection rolls differs significantly. The $RB$ state is composed of an equal weight superposition of two symmetry related oblique rolls $OR^{l}=\unicode[STIX]{x03C0}_{y}OR^{r}$. The $KN$ state in ILC at $\unicode[STIX]{x1D6FE}\neq 0^{\circ }$ is a mixed mode state, composed of transverse and longitudinal rolls that are not symmetry related and whose weight continuously changes along the branch. At $\unicode[STIX]{x1D6FE}=0^{\circ }$ however, $TR$ and $LR$ become symmetry related via rotation. The knot patterns observed in Rayleigh–Bénard convection (Busse & Whitehead Reference Busse and Whitehead1974) are thus expected to bifurcate in equivariant pitchfork bifurcations, like $RB$ in ILC.

3.5.1 Bifurcations to transverse oscillations

In all but one analysed parameter continuations, the pre-periodic orbit $TO$ bifurcates from $TR$ in a supercritical Hopf bifurcation. The bifurcations are either $\unicode[STIX]{x1D716}$-forward at $\unicode[STIX]{x1D716}\approx 0.07$, found at inclination angles $\unicode[STIX]{x1D6FE}=[80^{\circ },90^{\circ },100^{\circ },110^{\circ }]$, or $\unicode[STIX]{x1D6FE}$-backward, found at $\unicode[STIX]{x1D716}=0.1$ and $\unicode[STIX]{x1D6FE}=132.2^{\circ }$. The latter case represents the upper inclination limit of existence of $TO$ at $\unicode[STIX]{x1D716}=0.1$. At the lower limit, just below $\unicode[STIX]{x1D6FE}_{c2}$, $TO$ bifurcates as a quaternary state from the transverse subharmonic varicose state $TSV$ (see inset panel in figure 4). The $TSV$ state is an equilibrium state discussed briefly in appendix A. A common feature of all $TO$ branches is that the $\Vert \unicode[STIX]{x1D703}\Vert _{2}$-maximum over the orbit period remains close to the $\Vert \unicode[STIX]{x1D703}\Vert _{2}$-value of $TR$. This agrees with the observation that $TO$ modulations are sinusoidal oscillations around strictly transverse rolls with the maximum deflection associated to the minimum in $\Vert \unicode[STIX]{x1D703}\Vert _{2}$ over the orbit period (Reference Reetz and SchneiderRS20).

Figure 12.

Continuation of $TO$ at $\unicode[STIX]{x1D6FE}=100^{\circ }$ towards a chaotic state space. (a) Midplane temperature of $TO$ at $Ra=11\,000$ ($\unicode[STIX]{x1D716}=0.203$) before instabilities create a turbulent flow (b). The corresponding simulated time series of $TO$ (d) shows the transition from a periodic to a turbulent signal. The transition is a consequence of the many linear instabilities of $TO$ (f) that have emerged along the bifurcation branch (c). Solid/dotted bifurcation branches indicate stable and unstable states in the symmetry subspace of $S=\langle \unicode[STIX]{x03C0}_{y},\unicode[STIX]{x03C0}_{xz},\unicode[STIX]{x1D70F}(0.5,0.5)\rangle$. Here ‘RS20’ labels the control parameter value at which the temporal evolution has been studied in Reference Reetz and SchneiderRS20. The changing relative period $T^{\prime }$ of $TO$ along the bifurcation branch is shown in (e) and compared to the time scale of thermal diffusion $H^{2}/\unicode[STIX]{x1D705}$ and laminar mean advection $L_{x}/\bar{U}_{0}$ (dashed and dashed–dotted lines).

3.5.2 Numerical continuation of transverse oscillations

The $\unicode[STIX]{x1D6FE}$ continuation of $TO$ at $\unicode[STIX]{x1D716}=0.1$ is numerically straight forward and yields periodic orbits showing weak bending modulations around a purely transverse orientation (Reference Reetz and SchneiderRS20, § 4.2.2). All $\unicode[STIX]{x1D716}$ continuations are found to be numerically challenging for increasing $\unicode[STIX]{x1D716}$. We could not continue $TO$ much beyond $\unicode[STIX]{x1D716}=0.2$ (figure 12c). The reason for the computational difficulty is two-fold. First, the time period of the orbit drastically changes with $\unicode[STIX]{x1D716}$, which causes challenges for our shooting method. The pre-periodic orbit $TO$ satisfying (2.7) with $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70F}(0.5,0)$ oscillates slowly with a relative period $T^{\prime }\sim O(10^{2})$ close to the heat diffusion time $H^{2}/\unicode[STIX]{x1D705}$ and the laminar mean advection time $L_{x}/\bar{U}_{0}$. Along the continuation, the large orbit period is subject to significant and non-monotonic changes over small $\unicode[STIX]{x1D716}$-intervals (figure 12e). These changes in the orbit period are numerically difficult to trace. Secondly, the iterative solver of the Newton algorithm converges better if the target state is dynamically stable or weakly unstable (Sanchez et al. Reference Sanchez, Net, Garcıia-Archilla and Simo2004). Computing the spectrum of eigenvalues of $TO$ in the symmetry subspace of $[12\unicode[STIX]{x1D706}_{x},6\unicode[STIX]{x1D706}_{y}]$-periodicity indicates that the state branch at $\unicode[STIX]{x1D6FE}=100^{\circ }$ and $\unicode[STIX]{x1D716}=0.203$ has collected 63 unstable eigenvalues with a broad range of frequencies $\unicode[STIX]{x1D714}_{i}$ (figure 12f). At these parameters, the single-shooting Newton algorithm converged $TO$ to a residual of $\Vert {\mathcal{G}}(\boldsymbol{x})\Vert _{2}<3\times 10^{-11}$ (see (2.11)). When integrating the converged orbit forward in time, unstable directions trigger a transition to a turbulent state after $t=500$ (figure 12d). This turbulent state has been described as longitudinal bursts within switching diamond panes (Daniels et al. Reference Daniels, Plapp and Bodenschatz2000). We conclude that continuation of $TO$ for $\unicode[STIX]{x1D716}>0.2$ is challenging due to the numerical condition of a temporally slow, spatially large and very unstable periodic orbit that competes with many fast and small-scale modes in a chaotic turbulent state space.