Transition to turbulence in finite-dimensional approximations to the Boussinesq equations

Finite dimensional truncations were developed for the Boussinesq equations which govern the convective motion in a fluid layer heated from below. For a particular class of solutions to the truncate equations it was proven that the state of no convection is globally stable for all values of the Rayleigh number less than the critical Rayleigh number. A large class of truncations with bounded solutions was exhibited. Numerical techniques were used to study a particular nonlinear 14 component model for values of the Rayleigh number which exceed the critical value. By following the stationary solutions as a function of increasing Rayleigh number, that value of the Rayleigh number for which the stationary solutions become unstable was identified. When the solutions become unstable, there immediately follows a region of periodic motion, beyond which transition to turbulence occurs.