Modeling the atmospheric mixed layer

The atmospheric boundary layer, ABL, is the layer of air above the surface of the earth in which effects of the surface heating, cooling and friction are felt directly on time scales less than a day and in which significant fluxes of momentum, heat or matter are carried by turbulent motions on a scale of the order of the depth of the boundary layer. Over land, the ABL is strongly influenced by the diurnal cycle of surface heating and cooling. The unstably stratified ABL; or convective boundary layer, occurs when strong surface heating produces thermal instability. Stably stratified ABL occurs at night in response to surface cooling.; In this dissertation, we build a simple dynamical model for the atmospheric mixed layer leaving many details aside and concentrating instead on the most fundamental aspects such as heating from the ground, radiative heating and cooling, the effects of vertical shear, the earth's rotation, forcing and damping. Starting from conservation of mass, momentum and energy and a simple equation of state, we build a one-dimensional model of the atmosphere in the z-direction. We introduce a change of coordinates from (z, t) to (s, t) where s is the entropy, mainly to avoid any numerical diffusion. With an entropy appropriately defined, the model assigns a constant entropy value to the mixed layer, and above the mixed layer, the entropy decreases to 0 as z increases. We improve the model in every chapter; starting with an isothermal atmosphere, then changing the temperature profile to any function of z, and eventually letting the temperature evolve itself according to the equations. After that, we add vertical shear to the system. The shear has the effect that, when it crosses a threshold, it introduces instability to the system. With the addition of a shear in a second horizontal direction, we need to include the Earth's rotation, and then study the effects of resonance between the diurnal cycle and the Coriolis effect as well as forcing and damping terms. These cases are all implemented numerically.