Physica D
**191** (2004) 282-296.
10.1016/j.physd.2003.12.003
## Boundary effects and the onset of Taylor vortices

A.M. Rucklidge(1)
and
A.R. Champneys(2).

(1) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

(2) Department of Engineering Mathematics,

University of Bristol, Bristol, BS8 1TR, UK

**Abstract.**
It is well established that the onset of spatially periodic vortex states in
the Taylor-Couette flow between rotating cylinders occurs at the value of
Reynolds number predicted by local bifurcation theory. However, the symmetry
breaking induced by the top and bottom plates means that the true situation
should be a disconnected pitchfork. Indeed, experiments have shown that the
fold of the disconnected branch can occur at more than double the Reynolds
number of onset. This leads to an apparent contradiction: why should Taylor
vortices set in so sharply at the value Reynolds number predicted by the
symmetric theory, given such large symmetry-breaking effects caused by the
boundary conditions? This paper offers a generic explanation. The details are
worked out using a Swift-Hohenberg pattern formation model that shares the same
qualitative features as the Taylor-Couette flow. Onset occurs via a wall mode
whose exponential tail penetrates further into the bulk of the domain as the
driving parameter increases. In a large domain of length L, we show that the
wall mode creates significant amplitude in the centre at parameter values that
are O(L^{-2}) away from the value of onset in the problem with ideal boundary
conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in
space, which occurs at the same parameter value as the pitchfork bifurcation of
the temporal dynamics. The disconnected anomalous branch remains O(1) away from
the onset parameter since it does not arise as a bifurcation from the wall
mode.

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