Absolute and convective instabilities of, and signalling in, a flow in a porous medium with inclined temperature gradient and vertical throughflow

Flows in saturated porous media, such as ground water flows, saturated contaminant flows in a contaminanted soil and magma flows in the Earth's mantle are considerably influenced by an inhomogeneous temperature distribution in the media. In a tyipical model geometry of such flows, the porous medium is assumed to be a horizontal layer. The temperature of the bottom of the layer is supposed to be greater that that of its top resulting in the vertical variation of temperature within the layer. Also, according to observations one imposes a horizontal variation of temperature and allows for a vertical throughflow in the model. If strong enough, the variability of temperature can trigger a motion of the fluid deviating form the base state, i.e. a convection. Such a motion emerges as a consequense of the destabilisation of the basic state. The destabilisations occurs owing to the localised perturbations of the flow that are present under all conditions. The purpose of the proposed research is to use the modern methods of the linear stability theory in order to extend the existing treatments of convection in a porous medium with inclined temperature gradient, that analyse only spatially sinusoidal perturbations, to treating realistic localised disturbances. An analysis of localised disturbances would allow us to distinguish between two different destabilisation scenarios. In the first one the localised disturbances grow indefinitly at every location of the flow, thus destroying the base state throughout. This is the scenario of absolute instability. In the second case, the localised unstable perturbations move away from the place of their origin leaving behind an unperturbed state. Such a case is defined as the case of convective instability. In this case the base state flow, though being unstable, can be viewed as representing a physical end state in a certain portion of space. In other words, the absolute instability is a catastrophic instability as it results in the destruction of the base state throughout, whereas a convectively unstable, but absolutely stable state can be viewed as unperturbed in a certain portion of space despite the instability. This distinction means that the emergence of a secondary motion in the model and the characteristics of this motion depend on whether the unstable state is absolutely unstable or absolutely stable, but convectively unstable.In the proposed research, we first extend the existing analysis of monochromatic disturbances for a variety of the control parameter values (the vertical and horizontal Rayleigh numbers and the Peclet number) and obtain neutral curves. Further, localised disturbances will be treated by using the methods of the theory of absolute and convective instabilities. The numerical procedure in the treatment uses a Chebyshev collocation method for discretising the stability problem and a software implemention of the QZ-algorithm for treating the resulting generalised algebraic eigenvalue problem.