Normal and abnormal turbulent transport in a stratified medium

The influences of memory effects, coupling between velocity and temperature fluctuations and tensorial transport properties on momentum and heat transfers in turbulent flow which do not follow the Boussinesq relation are examined. It is shown that the memory effect, represented by the Lagrangian of the velocity gradients, can account for the decoupling between the flux and the gradient, while the tensorial properties of the transport coefficients allow a normal Boussinesq-type transfer with memory and anomalous counter-gradient or gradient-less transport.