The global dynamics of three-tori associated with homoclinic/heteroclinic global (infinite period) bifurcations are investigated for the magnetized spherical Couette problem, a paradigmatic model in geo- and astrophysical magnetohydrodynamics (MHD). A novel homoclinic bifurcation, involving collision between three-tori, is described. In addition, a heteroclinic bifurcation connecting two unstable two-tori with a stable three-torus is also analyzed. The role of the flow's spatial symmetries in this bifurcation scenario is also investigated. This bifurcation scenario gives rise to MHD flows that combine small with extremely large time scales.

The study of stable/unstable regular states in fluid systems is of paramount importance since they organize the dynamics by shaping the phase space (e.g., Ref. 1). Revealing the phase space geometry and the topology of the different attractors is, then, fundamental for explaining and forecasting global dynamics. With this aim, theoretical and experimental efforts have been made in the analysis of global bifurcations, i.e., sudden changes of the state topology occurring in the phase space when some input parameters are varied. One of these types of bifurcations is called “gluing” since, for the most simple case, there are two stable periodic orbits (PO) that glue together at the critical parameter to form a single PO. There are, however, few examples (e.g., Refs. 2 and 3) for which the glued states are invariant tori (quasiperiodic orbits).

As theoretically discussed in Ref. 4 on a simple model, the symmetries present in the system have consequences for gluing bifurcations, allowing collisions of multiple regular states rather than a single collision. This has been also confirmed in Ref. 2 for the cylindrical axisymmetric Taylor–Couette flow. In the case of weakly rotating Rayleigh–Bénard convection with an imposed magnetic field,5 gluing bifurcations of periodic orbits cease to exist due to the breaking of the rotation symmetry about the vertical axis.

By analyzing the spatial symmetries and the fundamental frequencies (FF) of the flow, we describe in this Letter a new gluing bifurcation scenario for the creation/destruction of three-tori in magnetohydrodynamic (MHD) systems, as is the case of the magnetized spherical Couette (MSC) problem. The present study constitutes the first example of global bifurcation phenomena occurring in a system capturing the essential features—rotation, spherical geometry, and magnetic fields—of geo- and astrophysical flows. The MSC problem has been largely considered, and it is still a problem of great interest, being investigated either experimentally (e.g., Ref. 6) or numerically,7 or both,8 since it represents a canonical model for the study of the planetary or stellar internal flow dynamics (e.g., Refs. 9 and 10). The analysis of these dynamics is essential, since dynamo action and magnetohydrodynamic (MHD) phenomena of celestial bodies are intrinsically coupled with the dynamics of conducting fluids in their interiors.

The MSC system is described by considering an electrically conducting fluid enclosed by an inner and outer spheres of radius r i and r o, respectively. A differential rotation is imposed by rotating the inner sphere at angular velocity Ω (around the vertical axis e z ̂) and keeping at rest the outer sphere. Finally, a homogeneous magnetic field, of modulus B0, in the axial direction is applied. If the fluid density is ρ, the kinematic viscosity is ν, and the magnetic diffusivity is η = 1 / ( σ μ 0 ) (μ0 is the magnetic permeability of the free-space, and σ is the electrical conductivity), the Navier–Stokes and induction equations modeling the system are
t v + Re ( v · ) v = p + 2 v + Ha 2 ( × b ) × e z ̂ , 0 = × ( v × e z ̂ ) + 2 b , · v = 0 , · b = 0.
The nondimensional numbers are defined as Re = Ω r i d / ν (Reynolds number) and Ha = B 0 d ( σ / ( ρ ν ) ) 1 / 2 (Hartmann number), and v and b represent the velocity field and the deviation of magnetic field from the applied field, respectively. To obtain the above-mentioned specific dimensionless form of the equations, length, time, velocity, and magnetic field are scaled with d = r o r i , d 2 / ν , r i Ω, and B0, respectively. In this study, the boundary conditions for v are no-slip ( v r = v θ = v φ = 0) at r = r o and constant rotation ( v r = v θ = 0 , v φ = sin θ, θ being colatitude) at r = r i. For the magnetic field, insulating boundary conditions are employed and the aspect ratio is fixed to χ = r i / r o = 0.5, following previous theoretical,11 as well as experimental studies.6 We notice that the inductionless approximation for small magnetic Reynolds number, Rm = Ω r i d / η 1, has been used to derive the equations. In our case, this is valid since we consider a moderate Re = 10 3 and a liquid metal (GaInSn) with magnetic Prandtl number Pm = ν / η O ( 10 6 ), which gives Rm = PmRe 10 3. The MSC equations are solved by means of spectral methods (spherical harmonics in the angular coordinates and a collocation method in the radial direction) and high order implicit–explicit backward-differentiation (IMEX–BDF) time schemes, which allow high-fidelity direct numerical simulations (DNS) to perform accurate time series analysis.

The MSC system admits an axisymmetric steady base state, symmetric with respect to the equator. Previous studies11 have shown that at moderate Re = 10 3 and small Ha < 12.2, the first instability takes the form of a rotating wave (RW) with threefold (m = 3) azimuthal symmetry. Two additional unstable RWs, with m = 2 or m = 4 azimuthal symmetry, are the origin of a rich dynamical phenomenon appearing as Ha is varied: symmetry breaking Hopf-type bifurcations of modulated rotating waves (MRWs) with two, three, and even four FF12 and intermittent behavior13 have been recently reported. In this study, we analyze stable MRW without any azimuthal nor equatorial symmetry for Ha [ 2.4 , 3.4 ]. The dependence of the volume-averaged nonaxisymmetric ( m 0) kinetic energy K na on Ha is displayed in Fig. 1. The labels 2Tm or 3Tm indicate the number of FF (2 or 3) and the azimuthal symmetry m = 1 or 2, which does not coincide with the largest energy component m = 4 (contour plots in Fig. 1). From initial conditions on the unstable branch 3T2-II (first described in Ref. 13), a 2T1-I MRW is obtained. By increasing Ha with small steps (down to 10 3), the modulation period undergoes a period doubling, and the branch 2T1-II is born. The notation I or II distinguishes this bifurcation. By further increasing Ha, a Hopf-type bifurcation occurs giving rise to the branch 3T1-II. On the latter, two global bifurcations (vertical lines in Fig. 1) have been detected.

FIG. 1.

Volume-averaged nonaxisymmetric kinetic energy density K na vs Ha. Its maximum, time average, and minimum correspond to the top, middle, and bottom curves, respectively. The MRWs, with different azimuthal symmetry m, are labeled 2Tm or 3Tm depending on whether the number of FF is 2 or 3. Solid/dashed lines represent stable/unstable waves. The position of the global bifurcations along the branch 3T1-II is given by vertical lines. Contour plots of the nonaxisymmetric radial velocity on an equatorial section at Ha = 3.25 are included.

FIG. 1.

Volume-averaged nonaxisymmetric kinetic energy density K na vs Ha. Its maximum, time average, and minimum correspond to the top, middle, and bottom curves, respectively. The MRWs, with different azimuthal symmetry m, are labeled 2Tm or 3Tm depending on whether the number of FF is 2 or 3. Solid/dashed lines represent stable/unstable waves. The position of the global bifurcations along the branch 3T1-II is given by vertical lines. Contour plots of the nonaxisymmetric radial velocity on an equatorial section at Ha = 3.25 are included.

Close modal

Gluing (infinite period) bifurcations of quasiperiodic states associated with homoclinic/heteroclinic behavior are characterized by a logarithmic dependence (e.g., Ref. 14) of a characteristic period vs the control parameter. In our case, the characteristic period is defined as T max = 1 / f min, where f min is the smallest frequency of the power spectral density of the time series of K4 (the volume-averaged kinetic energy contained in the m = 4 mode). Figure 2 confirms a piece-wise logarithmic dependence of T max (solid line) close to homoclinic (heteroclinic) bifurcations at Ha hom ( Ha het).

FIG. 2.

Maximal period T max vs the Hartmann number. Points represent computed solutions, and the solid lines the fitted curves. The errors of α, β, and γ are less than 3.5%.

FIG. 2.

Maximal period T max vs the Hartmann number. Points represent computed solutions, and the solid lines the fitted curves. The errors of α, β, and γ are less than 3.5%.

Close modal

For investigating the two bifurcations, the symmetries present in the flow are first described. This has been already addressed in Ref. 15, for the case of the purely hydrodynamic spherical Couette. When the equatorial symmetry of the base state is broken at the first bifurcation, the emerging RWs still conserve a shift-and-reflect symmetry T, which depends on the azimuthal wave number m of the RW. In the case of the MSC, the situation is the same, since for the selected Ha, the first and subsequent instabilities are equatorially antisymmetric. For the concrete value of m = 4, the symmetry transformation T corresponds to a reflection with respect to the equator and an azimuthal rotation of π / 4. Successive bifurcations with possible azimuthal symmetry breaking, originated from RW with m = 4, give rise to MRW described by Garcia et al.13 for which the T-symmetry may be broken or not. To check the latter, DNS of T-transformed initial conditions have been performed.

To elucidate which regular flows take part in the two bifurcations, Poincaré sections defined in terms of the velocity field are plotted in Fig. 3. The two Poincaré sections of a 3T1-II MRW (black), and those of the corresponding T-transformed MRW (blue), are displayed on panel (a) for Ha Ha hom. The banded loops on that figure are inherited from the period doubling bifurcation giving rise to 2T1-II (see Fig. 1). At Ha Ha hom, the corresponding flows are T-invariant and their Poincaré sections [see panel (b) of Fig. 3] are basically the union of the sections shown in panel (a). In addition, it seems that T-invariant unstable MRW on the 3T2-II branch is also playing a role on this bifurcation. The two sections of the rightmost solution of the 3T2-II branch in Fig. 1 are also shown (in red) in Fig. 3(a). By a close inspection, a part of these two sections can be identified in Fig. 3(b). Notice that this identification is not so clear since 3T2-II can be only computed up to Ha = 2.9, which is not so close to Ha hom. These unstable 3T2-II seem to be also involved in the second global bifurcation at Ha = Ha het as evidenced in Fig. 3(c). For Ha Ha het, the latter figure displays points that are located on the same region as some of the points [red, Fig. 3(a)] of the unstable 3T2-II. However, the section of Fig. 3(c) no longer displays the banded loops corresponding to the period doubling (branch 3T1-II), but it is clearly related to the 2T1-I and its T-transformed branch, since the Poincaré sections (not shown in the figure) are quite similar.

FIG. 3.

Poincaré sections defined by v r ( r 1 , 3 π / 8 , π / 2 ) = 5. × 10 2. The radial positions are r 1 = r i + 0.5 d and r 2 = r i + 0.85 d. (a) The two Poincaré sections of 3T1-II at Ha = 3.236 (black) and the corresponding T-transformed MRW (blue). The two sections of a T-invariant unstable 3T2-II (at Ha = 2.9) are also shown (red). The T transformation is a reflection with respect to the equator and an azimuthal rotation of π / 4. (b) and (c) The two Poincaré sections of the T-invariant 3T1-II at Ha = 3.24 and Ha = 3.268, respectively.

FIG. 3.

Poincaré sections defined by v r ( r 1 , 3 π / 8 , π / 2 ) = 5. × 10 2. The radial positions are r 1 = r i + 0.5 d and r 2 = r i + 0.85 d. (a) The two Poincaré sections of 3T1-II at Ha = 3.236 (black) and the corresponding T-transformed MRW (blue). The two sections of a T-invariant unstable 3T2-II (at Ha = 2.9) are also shown (red). The T transformation is a reflection with respect to the equator and an azimuthal rotation of π / 4. (b) and (c) The two Poincaré sections of the T-invariant 3T1-II at Ha = 3.24 and Ha = 3.268, respectively.

Close modal

A brief schematic diagram summarizing the flow bifurcation phenomena is shown in Fig. 4. The different branches of MRW involved in the bifurcations are displayed. On this scheme, the T-symmetry corresponds to a reflection with respect to the horizontal line. Thus, T-invariant solutions lie on the horizontal axis. At Ha = Ha hom, two stable T-symmetry related 3T1-II collide giving rise to the T-invariant stable 3T1-II. The latter solutions lie close to unstable 2T1-II as evidenced by the Poincaré sections of Fig. 3. As Ha is further increased, unstable 2T1-I also approaches the 3T1-II branch. From this point, we assume that there is multiple heteroclinic collisions between the 2T1-I and 2T1-II unstable branches at Ha = Ha hom giving rise to the rest of the 3T1-II. This is reasonable since for Ha < Ha het, the Poincaré sections of 3T1-II relate to those of 2T1-II, whereas for Ha > Ha het, the Poincaré sections of 3T1-II relate to those of 2T1-I. To provide further evidence that this may be indeed the case, the time series of volume-averaged kinetic energies of the modes m = 1 , 2 , 3 , 4, for a 3T1-II at Ha = 3.25, are displayed in Fig. 5(a). In this figure, two different dynamical behaviors (marked with a vertical line) can be identified. The first (around t = 5) is represented by a (roughly) periodic oscillation lasting several time units, almost matching that observed for solutions on the 2T1-II branch. In contrast, the second dynamical behavior occurs quite suddenly and seems to be related to the 3T2-II branch. This is evidenced in the contour plots shown in the bottom part of Fig. 5 (see figure caption), where the snapshots of a 3T2-II (at Ha = 2.6) and those of a 2T1-II (top left and right, respectively), and the corresponding snapshots of a 3T1-II (at Ha = 3.25) at the time instants marked with vertical lines (bottom left and right, respectively), compare quite well.

FIG. 4.

Scheme of the bifurcation sequence for invariant tori u and its symmetry related partner T ( u ) vs Ha. The branches on the horizontal axis are T-invariant. Solid/dashed lines represent stable/unstable branches. The different branches of modulated rotating waves are labeled 2Tm or 3Tm, according to their number (2 or 3) of FF. The subscript m represents the azimuthal symmetry. The colors distinguish the different branches.

FIG. 4.

Scheme of the bifurcation sequence for invariant tori u and its symmetry related partner T ( u ) vs Ha. The branches on the horizontal axis are T-invariant. Solid/dashed lines represent stable/unstable branches. The different branches of modulated rotating waves are labeled 2Tm or 3Tm, according to their number (2 or 3) of FF. The subscript m represents the azimuthal symmetry. The colors distinguish the different branches.

Close modal
FIG. 5.

Top: Volume-averaged kinetic energies of the modes m = 1 , 2 , 3 , 4 vs time for 3T1-II at Ha = 3.25. Bottom: Contour plots of the nonaxisymmetric component of the radial velocity on a section at θ 93 °. The snapshot on the top left corresponds to 3T2-II at Ha = 2.6, whereas the snapshot on the top right corresponds to 2T1-II at Ha = 3.23. The two snapshots on the bottom (at t = 5 on the left and at t = 25 on the right, see vertical lines on the top figure) correspond to 3T1-II at Ha = 3.25.

FIG. 5.

Top: Volume-averaged kinetic energies of the modes m = 1 , 2 , 3 , 4 vs time for 3T1-II at Ha = 3.25. Bottom: Contour plots of the nonaxisymmetric component of the radial velocity on a section at θ 93 °. The snapshot on the top left corresponds to 3T2-II at Ha = 2.6, whereas the snapshot on the top right corresponds to 2T1-II at Ha = 3.23. The two snapshots on the bottom (at t = 5 on the left and at t = 25 on the right, see vertical lines on the top figure) correspond to 3T1-II at Ha = 3.25.

Close modal

A complex novel bifurcation scenario, involving collisions among two- and three-tori and giving rise to infinite-period flows, has been analyzed in this study. The azimuthal and equatorial symmetries of all stable flows are totally broken, but a shift-and-reflect symmetry plays a key role at the two bifurcations. For the first, two branches of T-symmetry related three-tori merge onto a branch of three-tori, which are T-invariant. At this point, an homoclinic connection with an unstable three-torus (m = 2 azimuthally symmetric) seems to occur. The second bifurcation appears to be related to heteroclinic behavior associated with branches of unstable two-tori, which have lost all spatial symmetries. The study of high-dimensional tori, in symmetric fluid systems (e.g., Ref. 12), is of fundamental importance since these solutions are the origin of chaotic and turbulent behavior, commonly found in scientific and engineering applications. For instance, homoclinic/heteroclinic phenomena are associated with coherent states at the edge of turbulent shear flows (e.g., Refs. 16 and 17), and thus, their study is essential for understanding the transition to turbulence. The application of dynamical systems concepts to large-scale nonlinear systems is computationally challenging,18 but brings out significant advances in their understanding. In the case of spherical fluid systems, modeling multitude of geo- and astrophysical phenomena, high-dimensional tori take the form of MRW, which are the origin of triadic resonant motions observed in chaotic flows.19 The present study is, thus, also important in the field of planetary and stellar flows, since it evidences a bifurcation mechanism giving rise to very large-scale temporal oscillations on MHD systems.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 787544).

The author has no conflicts to disclose.

Fernando Garcia: Conceptualization (equal); Data curation (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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