Andrew Soward | Global bifurcation to travelling waves with application to narrow
gap spherical Couette flow |

Abstract:

In a previous paper (Harris *et al*, 2000, Physica D **137**,
260-276, [1]), an inhomogeneous complex Landau equation was derived in
the context of the amplitude modulation of Taylor vortices between two
rapidly rotating concentric spheres, which bound a narrow gap and almost
co-rotate about a common axis of symmetry. In this weakly nonlinear regime
the latitudinal vortex width is comparable to the gap between the shells.
The vortices are located close to the equator and are modulated on a latitudinal
length scale large compared to the gap width but small compared to the
shell radius.

Here we investigate both analytically and numerically the stability
and subsequent evolution of the steady finite amplitude solutions found
in [1]. A complicated bifurcation structure is unravelled dependent on
the magnitude *k* of spatial phase mixing. Only when the inner and
outer spheres almost corotate is *k* small; otherwise *k*
is large. Two types of mode exist-- one (SP) preserves the reflectional
symmetry of the steady solutions with respect to the equatorial plane while
the other type (SB) breaks it. For sufficiently large *k*, a supercritical
SP-Hopf bifurcation of the steady state leads to a vacillating solution
which expands into a homoclinic cycle connecting the trivial undisturbed
state to itself. SP-global bifurcations occur for all *k* leading
to limit cycles which correspond to vortices drifting towards the equator
from both sides. For small *k* a pair of heteroclinic connections
is made between two steady states of equal amplitude but of opposite sign.
As *k* is increased the steady state amplitude falls to zero leading
for large *k* to the gluing of the aforementioned homoclinic cycles.
For moderate *k* the nonlinear development shows no evidence of any
stable symmetry-broken temporally periodic states although for large phase
mixing *k >> 1* strongly subcritical finite amplitude periodic SB-solutions
are identified. On increasing the driving parameter (related to the Taylor
number) at fixed large *k* both SP- and SB-periodic travelling wave
solutions are obtained numerically. The asymmetric SB-waves correspond
to vortices drifting across the equator although far from it, where these
vortices are very weak, they drift towards the equator as in the case of
the SP-waves.

This is joint work with

Derek Harris^{1} (D.Harris@ex.ac.uk),

and Andrew P. Bassom^{1,2} (A.P.Bassom@ex.ac.uk).

^{1} *School of Mathematical Sciences, University of
Exeter, Exeter, EX4 4QE, UK.*
^{2} *School of Mathematics, University of New South
Wales, Sydney 2052, Australia.*