Linear mechanism of surface gravity wave generation in horizontally sheared flow

An analysis is presented of a linear mechanism of surface gravity wave generation in a horizontally sheared flow in a fluid layer with free boundary. A free-surface flow of this type is found to be algebraically unstable. The development of instability leads to the formation of surface gravity waves whose amplitude grows with time according to a power law. Flow stability is analyzed by using a nonmodal approach in which the behavior of a spatial Fourier harmonic of a disturbance is considered in a semi-Lagrangian frame of reference moving with the flow. Shear-flow disturbances are divided into two classes (wave and vortex disturbances) depending on the value of potential vorticity. It is shown that vortex disturbances decay with time while the energy of wave disturbances increases indefinitely. Transformation of vortex disturbances into wave ones under strong shear is described.