Nonlinearity **9** (1996) 311-351.
doi:10.1088/0951-7715/9/2/003
## Analysis of the shearing instability in nonlinear convection and
magnetoconvection

A.M.Rucklidge and
P.C. Matthews

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, CB3 9EW, UK

**Abstract.**
Numerical experiments on two-dimensional convection with or without a vertical
magnetic field reveal a bewildering variety of periodic and aperiodic
oscillations. Steady rolls can develop a shearing instability,
in which rolls turning over in one direction grow at the expense of rolls
turning over in the other, resulting in a net shear across the layer. As the
temperature difference across the fluid is increased, two-dimensional pulsating
waves occur, in which the direction of shear alternates. We analyse the
nonlinear dynamics of this behaviour by first constructing appropriate
low-order sets of ordinary differential equations, which show the same
behaviour, and then analysing the global bifurcations that lead to these
oscillations by constructing one-dimensional return maps. We compare the
behaviour of the partial differential equations, the models and the maps in
systematic two-parameter studies of both the magnetic and the non-magnetic
cases, emphasising how the symmetries of periodic solutions change as a result
of global bifurcations. Much of the interesting behaviour is associated with a
discontinuous change in the leading direction of a fixed point at a global
bifurcation; this change occurs when the magnetic field is introduced.

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