  While a theoretical limit has long been established for the performance of a
single turbine, no corresponding upper bound exists for the power output from a
large wind farm, making it difficult to evaluate the available potential for
further performance gains. Here we build a model describing the essential
features of a large array of turbines with arbitrary design and layout, by
considering a fully-developed wind farm whose upper edge is bounded by a
self-similar boundary layer. The exchanges between the wind farm, the
overlaying boundary layer, and the outer flow are parameterized by means of the
classical entrainment hypothesis. We obtain a concise expression for the wind
farm's power density (corresponding to power output per unit planform area), as
a function of three coefficients, which represent the array thrust and the
turbulent exchanges at each of the two interfaces. We assess the performance of
our model by comparing the predicted power density to field data, laboratory
measurements and large-eddy simulations for the fully-developed regions of wind
farms, finding good agreement. We extend our model to include the effect of
atmospheric stability on power output, by using an established parameterization
relating entrainment coefficients to local Froude numbers. Our predictions for
power variation with atmospheric stability are in broad agreement with field
measurements. To the best of our knowledge, this constitutes the first
quantitative comparison between an atmospheric-stability-dependent model and
field data. Finally, we consider an ideal limit for array operation, whereby
turbines are designed to maximize momentum exchange with the overlying boundary
layer. This enables us to obtain an upper bound for the performance of large
wind farms, which we determine to be an order of magnitude larger than the
output of contemporary turbine arrays.
