Stability of algebraic non-equilibrium second-order closure models

Abstract

Stability problems of algebraic non-equilibrium second-moment closure models have given rise to the so-called quasi-equilibrium versions in which turbulence equilibrium is used as an additional constraint. In this paper, we investigate reasons for the failure of the G.L. Mellor, T. Yamada [Reviews of Geophysics 20 (1982) 851] level 2.5 closure model and suggest a remedy for this. We further discuss a new non-equilibrium closure model by V.M. Canuto, A. Howard, Y. Cheng, M.S. Dubovikov (Journal of Physical Oceanography, 2000, accepted for publication) which has proven to allow for stable calculations. All models are then numerically tested with a simple wind entrainment experiment motivated by the H. Kato, O.M. Phillips [Journal of Fluid Mechanics 37 (1969) 643] laboratory experiment, with the aid of which the instability of the Mellor and Yamada (1982) and the stability of the Canuto et al. (2000) model are confirmed. The Canuto et al. (2000) model has three advantages compared to the Mellor and Yamada (1982) which are (i) the symmetry of stability functions, (ii) a higher critical Richardson number, and (iii) that the normalised shear stress increases with normalised shear for turbulence equilibrium. The latter advantage of the new model causes its high physical and numerical stability.