In a multiscale system unresolved scales affect not only the dynamics but also the statistics of the large scales. In particular, small scales in atmospheric turbulence are responsible for the dissipation of kinetic energy from larger scales. This cascade of energy establishes an equilibrium kinetic energy spectrum. Because of the level of certainty in these statistical observations we consider statistical accuracy as our primary measure for model performance.
In this light we study the viscous Burgers' equation as a model for atmospheric turbulence and add a control to enforce the correct spectrum. The control has a form originally proposed by Leimkuhler, Legoll and Noorizadeh (2008) for controlling the kinetic energy in molecular dynamics simulations. This control has the desirable property that its effects weaken as the solution is closer to equilibrium.