A numerical study of nonlinear cascading of atmospheric baroclinic and barotropic flow with a two-layer quasi-geostrophic model
The conversion and nonlinear triad exchanges of energy is investigated in a quasi-geostrophic two-layer channel atmospheric model. The two vertical (barotropic and baroclinic) and longitudinal (Fourier) normal modes are used to expand the quasi-geostrophic energy and potential enstrophy equations. The quasi-geostrophic zonal and eddy (and zonal wave number) barotrophic and baroclinic equations are noted to be different than those derived previously (Wiin-Nielsen and Drake, Mon. Wea. Rev. 94:221, 1966; Chen and Tribbia, Dept. Earth Sci., ISU, 1981) for the primitive equation model. These differences are used to explain some previous primitive equation calculations. Because of these differences, some conversions are calculated for the first time;Five experiments were performed based on expectations from linear baroclinic instability theory and inferred finite amplitude behavior. Some expectations from linear baroclinic instability theory are not found. Reasons for this failure are proposed, including the barotropic-baroclinic triad behavior (Marshall and Chen, Geophys. Astrophys. Fluid Dyn. 22:21, 1982). Increases in friction produce steeper power laws of energy and potential enstrophy. The cascades for these different power laws are compared;A three-dimensional index is formed by the decomposition of kinetic energy into barotropic and baroclinic vertical subdivisions and into meridional and zonal wave numbers. It is argued that the spectra of all energies and enstrophies, for this two-level model, can be inferred from the three-dimensional index of kinetic Oil, Gas, and Energy;The schematic cascading diagram of Salmon (Geophys. Astrophys. Fluid Dyn., 10:25, 1978), which is based on related oceanic cascading theory of Rhines (In The Sea, John Wiley, 1977), for the two-layer model is evaluated by a direct calculation in the zonal wave number index. Some modifications in the diagram of Salmon are found necessary from derivations and calculations. The tendency of the cascades to redistribute energy away from the source wave numbers is found to overpower other previously noted tendencies found in initialization experiments (Rhines, 1977). The conversion from baroclinic to barotropic energy occurs not only between the same wave number but between different ones as well, which results (from calculation) in a net upscale cascade of energy. A modified barotropic and baroclinic cascading diagram is proposed. These modifications may also relate to the original ocean cascading theory of Rhines.