Coriolis recovery of wind farm wakes

Abstract

Two mechanisms cause wind speed recovery in the wake of a wind farm: momentum mixing and the Coriolis force. To study these mechanisms, we use a steady linearised two-layer fast Fourier transform (FFT) model so that both analytical expressions and full flow fields can be derived. The model represents the complex vertical mixing processes by a simple Rayleigh friction. Rayleigh friction decays the wind disturbance at a rate proportional to its local value. Pressure gradient forces are computed using a two-part vertical wave number formulation in the upper layer. The Coriolis force recovery occurs by deflecting flow leftward (in the Northern Hemisphere). The Coriolis force, acting on this cross-flow, re-accelerates the flow in the downwind direction. The relative importance of Rayleigh versus Coriolis wake recovery depends on their two coefficients: C and f respectively, each with units of inverse time. When the coefficient ratio is large, i.e. C/f 1, Rayleigh friction restores the wake before Coriolis can act. Farm size and atmospheric static stability are also important in wake recovery. The wakes of small- and medium-size farms will quickly approach geostrophic balance. Once balance is established, the ratio of farm size “a” to the Rossby radius of deformation (RRD) determines the amount of Coriolis recovery. For a small farm in a stable atmosphere (a < RRD), Coriolis acts by adjusting the pressure field to obtain geostrophic balance rather than accelerating the wind. When this occurs, only momentum mixing can restore the “inner” wake. For large farms in less stable conditions (a > RRD), the Coriolis force significantly contributes to wake recovery. In this case, the leftward deflected flow creates “edge jets” on either side of the wake. Including the Coriolis force when modelling wind farm wakes is demonstrated to be increasingly important for larger wind farms or farm clusters.