Dynamic roughness and the transition between wind wave regimes

Abstract

A variational principle applicable to a closed and dissipative mechanical system is applied to wind waves. The result of this principle is that the water surface will take the form that provides the greatest surface drag and maximizes surface drag coefficient, or dynamic roughness. Therefore the normal train of logic is reversed: a maximum dynamic roughness is the criterion for gravity waves, capillary waves, or a hydraulically smooth water surface. Equations are assembled for the dynamic roughness of the water surface, and the variational principle is applied to find the transition between gravity and capillary waves and between capillary waves and a smooth water surface. It is found that wind fetch has no effect upon the point of transition from capillary to gravity waves. The transition between a smooth water surface and capillary waves, however, was found to depend upon both wind shear and fetch. The shear velocity of the transition increases with increasing fetch.