A new wind shear relation is derived based on the atmospheric equations of motion, which are, in turn, based on the Navier-Stokes equations. In order to make the obtained system of equations solvable, it is assumed that the air density and pressure follow the International Standard Atmosphere relations and that the wind speed is constant over time. Additionally, it is assumed that the gravity force is constant, that the air is incompressible and that there is no wind speed variation in x- and y-direction.
If the additional assumption of no vertical wind speed is made, then the relations simplify to the Ekman equations of motion, which result in the Ekman spiral. Not making this additional assumption results in a system of equations that can be solved exactly or using asymptotic matching. As expected, the solution that was derived using asymptotic matching approaches the exact solution.
The derived relation is compared to wind speed relations from literature, such as the log law, the profile by Gryning, the profile by Peña and the profile by Kelly. It appears that the log law performs best in terms of simplicity and accuracy, but a final conclusion cannot yet be drawn. Additional research is recommended into what types of locations can be accurately represented by the derived relation and into what input data is required and obtainable in order to increase the accuracy of the wind speed predictions by the theoretical profiles.