Limits to the power density of very large wind farms

A simple analysis is presented concerning an upper limit of the power density (power per unit land area) of a very large wind farm located at the bottom of a fully developed boundary layer. The analysis suggests that the limit of the power density is about 0.38 times $\tau_{w0}U_{F0}$, where $\tau_{w0}$ is the natural shear stress on the ground (that is observed before constructing the wind farm) and $U_{F0}$ is the natural or undisturbed wind speed averaged across the height of the farm to be constructed. Importantly, this implies that the maximum extractable power from such a very large wind farm will not be proportional to the cubic of the wind speed at the farm height, or even the farm height itself, but be proportional to $U_{F0}$.