When a deep body of fluid is heated from below at a horizontal boundary a destabilizing temperature gradient develops. As the heat diffuses into the fluid the effective thermal Rayleigh number based on the diffusion length-scale grows and, for the case of a sudden increase in the temperature by a fixed amount, is proportional to $t^{3/2}$. Hence,one may expect the fluid to be stable initially, only becoming unstable after a finite time. When an evolving thermal boundary layer first becomes unstable the growth rate of the instabilities may be comparable to the time-scale of the evolution of the background temperature profile, and so analytical approximations such as the quasi-static approximation (assuming the time-evolution of the background state can be ignored) are not strictly appropriate.
We will focus on three basic problems: heating a body of fluid from a horizontal boundary, heating from a vertical boundary and heating a
body of fluid with a stabilizing salinity gradient from below. We look at the growing phase of the linear instabilities as an initial value
problem where the initial time for the instabilities is a parameter to be determined. We will use a more general measure of the amplitude or
“energy” of the disturbances than, say, just the kinetic energy. We will determine numerically the optimal initial conditions and the optimal
starting time for the instabilities to ensure a given growth occurs at the earliest time for both heating from horizontal and vertical
boundaries. In some cases there is a clear change in form of the instabilities as they evolve.