Energy similarity—a new turbulence closure model for stable boundary layers (formally paper number J2.8)

We present a new turbulence closure model for stratified turbulence. The closure of the equation system, the equations for the mean flow, mean potential temperature, turbulent kinetic energy and potential temperature variation, is based on the energy similarity assumptions. These are that the non-dimensional fluxes of momentum and potential temperature can be described by single-valued functions of the local flow stability. These functions are taken from new results of renormalization group theory and we show that they agree well with observations. This theory includes the effect of turbulence anisotropy and local gravity wave viscosity. The non-dimensional flux of momentum is the stress divided by turbulent kinetic energy, while the non-dimensional potential temperature flux is the flux divided by the square root of turbulent kinetic energy times the temperature variance, respectively. The model requires a dissipation length-scale to be defined.

The model is tested for the first GABLS case. The simulation starts with a neutral boundary layer capped with an inversion and then surface cooling is applied. A constant cooling rate is applied to the surface, resulting in a weakly stable, quasi-stationary boundary layer about 150-200 meters deep. The results from the new turbulence model compares well with large eddy simulation results for the same case.

In the hierarchy of models, this energy similarity closure model is situated between the 1- 1.5-order closures, retaining either none or one auxiliary equation for turbulence, and the second order models that utilize 10 extra equations for all the second order moments of the flow. By using two equations for describing the turbulence field, we avoid the use of a mixing-length scale, and allow third-order flux divergences to affect the turbulence budgets in fundamentally different ways for momentum and heat. This is the case for gravity waves, which are generated in the boundary layer by the turbulence and emitted to the free troposphere. This is possible in the long-lived stable boundary layer found during winter and in polar and coastal regions, as shown in previous studies by one of the authors.