The nonlinear development of finite amplitude disturbances in mixed convection flow in a heated vertical annulus is studied by direct numerical simulation. The unsteady Navier-Stokes equations are solved numerically by a spectral method for different initial conditions. The results indicate that the equilibrium state of the flow is not unique, but depends on the amplitude and wavenumber of the initial disturbance. In all cases, the equilibrium state consists of a single dominant mode with the wavenumber k(f), and its superharmonics. The range of equilibrium wavenumbers k(f) was found to be narrower than the span of the neutral curve from linear theory. Flows with wavenumbers outside this range, but within the unstable region of linear theory, are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that linearly stable long and short waves can also excite a wave inside this narrow band through nonlinear wave interaction. The results suggest that the selection of the equilibrium wavenumber k(f) is due to a nonlinear energy transfer process which is sensitive to initial conditions. The consequence of the existence of nonunique equilibrium states is that the Nusselt number cannot always be expressed uniquely as a function of appropriate dimensionless parameters such as the Reynolds, Prandtl and Rayleigh numbers. Any physical quantity transported by the fluid, such as heat and salt, can at best be determined within the limit of uncertainly associated with nonuniqueness. This uncertainty should be taken into account when using any accurately measured values of the heat transfer rate since it is only one of the many possible states for the controllable conditions and geometry. Extrapolating this fact to turbulence, it is our opinion, since the time average will depend on the initial condition, it will not equal to the ensemble average even for stationary turbulence. This is because that the mean flow is not unique for a given Reynolds or Rayleigh number. Consequently, the ergodic hypothesis is not valid. From an application point of view, only the time average has physical significance.