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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