### Stability of explicit models

A physically reasonable condition for an explicit second order model expressed the fact that increasing (non-dimensional) shear should lead to increasing vertical shear-anisotropies of turbulence, and . It has been shown by Burchard and Deleersnijder (2001) that a violation of this condition may lead to numerical instabilities of the models.

Mathematically, the shear-condition is expressed by

 (82)

where (74) has been used. Using the equilibrium form of the stability function described in section 4.7.39, this condition leads to a cubic equation in . A simpler condition can be obtained, when this equation is solved after terms multiplied by and , which usually are very small, are neglected.

The resulting approximate condition is

 (83)

Burchard and Deleersnijder (2001) showed that using (83) the most well-known models yield numerically stable results. However, for some models like those of Mellor and Yamada (1982) and Kantha and Clayson (1994), the limiter (83) is almost always `active', and hence replaces the actual turbulence model in a questionable way.

Karsten Bolding 2012-01-24