Heteroclinic bifurcations in a simple model of double-diffusive convection

Two-dimensional thermosolutal convection is perhaps the simplest example of an idealized fluid dynamical system that displays a rich variety of dynamical behaviour which is amenable to investigation by a combination of analytical and numerical techniques. The transition to chaos found in numerical experiments can be related to behaviour near a multiple bifurcation of codimension three. The resulting third-order normal form equations provide a rational approximation to the governing partial differential equations and thereby confirm that temporal chaos is present in thermosolutal convection. The complex dynamics is associated with a heteroclinic orbit in phase space linking a pair of saddle-foci with eigenvalues satisfying Shil'nikov's criterion. The same bifurcation structure occurs in a truncated fifth-order model and numerical experiments confirm that similar behaviour extends to a significant region of parameter space.