Energy and enstrophy conservation properties of high-order non-oscillatory advection schemes

Abstract

The dissipative behaviour as well as the conservative and spectral properties of a class of high-order non-oscillatory advection schemes, developed by Holm (1995), are investigated for the incompressible, two-dimensional, constant density Euler equations. These equations are closely connected to large scale atmospheric flows. The advection schemes are applied to two different forms of the governing equations: the vorticity-stream function form and the momentum-pressure form. For both forms, the advection schemes are found to be essentially kinetic energy conserving. This includes several tests, for example isotropic turbulence simulations. Furthermore, enstrophy is dissipated at a correct rate predicted by theory. In turbulence simulations, enstrophy spectra with an inertial subrange slope of ∼ − 1.2 are observed, in accordance with theory. The time-evolution of the kinetic energy, enstrophy and the enstrophy dissipation are also found to be consistent with the theory of viscous, incompressible, two-dimensional turbulence. For the shorter waves, the spectral amplitudes fall off more rapidly than what is expected for the real flow. It is shown that this damping of the short waves is necessary for obtaining non-oscillatory physical fields, since otherwise oscillations would appear due to Gibbs' phenomenon. Thus it is impossible to have a perfect spectrum and a non-oscillatory physical field at the same time, and a compromise between these two aspects needs to be found. If it is desirable that the physical field be non-oscillatory, then the high-order, non-oscillatory schemes give better conservation properties and a more realistic spectrum than a reference energy and enstrophy conserving scheme supplied with artificial viscosity. In general, it is found that high-order, non-oscillatory advection schemes have the properties required for accurate simulations of large-scale atmospheric motions. DOI: 10.1034/j.1600-0870.1996.00007.x