Surface Roughness Height Model for Stratified Atmospheric Boundary Layer Interacting with Wind Farms

The interaction between the atmospheric boundary layer (ABL) in neutral conditions and fully-developed wind-turbine arrays has been recently investigated using large eddy simulation (LES). These studies resulted in derivations of new model equations for the effective surface roughness height, taking into account characteristics of the wind farm. In this work, a similar LES analysis is performed except the interaction of the thermally-stratified ABL with a fully-developed wind-turbine array is considered. This problem is relevant to both onshore environments, where during the day the ABL is unstable while during the night is stable, and offshore environments, where the ABL is most of the times stable. New model equations for the effective surface roughness height in thermally-stratified conditions are derived by considering, as in neutral conditions, two additional logarithmic mean velocity profiles, one within the rotor wakes and the other above the wind turbines, respectively. The effect of the rotor wakes is taken into account through an additional wake eddy viscosity coefficient, and an additional eddy diffusivity coefficient. The model consists of four coupled nonlinear equations solving for the effective roughness height, the friction velocities corresponding to the regions below and above the rotor, and the surface heat flux. LES results consisting of horizontally-averaged velocity, potential temperature, and vertical turbulent momentum and heat flux profiles are used to validate the model equations, and to characterize the impact of the fully-developed wind-turbine array on the thermally-stratified ABL (Authors acknowledge funding from the NSF, grant number AGS- 1045189).