Higher Order Nonlinear Schr(?)dinger Equation And Nonlinear Rossby Wave Packet With Complete Coriolis

Rossby is one of important waves,many invesgaters acheived certain acheivements on them.Based on the quasi-geostrophic potential vorticity equation with complete Coriolis force,the evolution of the nonlinear Rossby waves including the two contents of this paper:On the one hand,the evelution of the amplitude of Rossby solitary wave envelope includes complete Coriolis force satisfies the inhomogeneous nonlinear Schr(?)dinger equation by employing the perturbation method and stretching transforms of time and space.The results show the horizontal component of the complete Coriolis forrce has some effects on the Rossby solitary waves,at the same time,beta effect and topographic effect are also important to the Rossby solitary waves.If fH=0,the equation can be reduced to the results of former in special cases.On the other hand,a new perturbation expansion method is proposed to derive a higher nonlinear Schr(?)dinger equation(HNLS)including higher order terms to O(?)and O(?2)under a complete Coriolis force for weakly nonlinear Rossby waves.Based on this equation,the modulational instability of a uniform Rossby wave is discussed.It is found that the higher order terms to O(?)and O(?2)under the Coriolis force have an important effect on the modulational instability of a uniform Rossby wave train.At the same time,the area will become smaller with the basic flow.In addition,the uniform background basic flow does affect the modulational instability of uniform Rossby wave train.Similiarly,if ?= 0,the equation also can be reduced to the results of former in special cases.