Energy-conserving discretisation of turbulent shear and buoyancy production

An energy-conserving discretisation of the shear and buoyancy production terms for turbulent kinetic energy is derived. In contrast to `ad hocʹ numerical schemes, this guarantees that all mean kinetic and potential energy which is lost or gained due to vertical mixing is exactly subtracted or added to the turbulent kinetic energy budget. It is further shown in a numerical wind-entrainment experiment that this new methods results in significantly more stable numerical solutions.