  Three-dimensional double-diffusive convection in a horizontally infinite
layer of an uncompressible fluid interacting with horizontal vorticity field is
considered in the neighborhood of Hopf bifurcation points. A family of
amplitude equations for variations of convective cells amplitude is derived by
multiple-scaled method. Shape of the cells is given as a superposition of a
finite number of convective rolls with different wave vectors. For numerical
simulation of the obtained systems of amplitude equations a few numerical
schemes based on modern ETD (exponential time differencing) pseudo-spectral
methods were developed. The software packages were written for simulation of
roll-type convection and convection with square and hexagonal type cells.
Numerical simulation has showed that the convection takes the form of elongated
"clouds", "spots" or "filaments". It was noted that in the system quite rapidly
a state of diffusive chaos is developed, where the initial symmetric state is
destroyed and the convection becomes irregular both in space and time. The
obtained results may be the basis for the construction of more advanced models
of multi-component convection, for instance, model of Saturn's polar hexagon.
