Properties and stability analysis of the sixth-order Boussinesq equations for Rossby waves

Abstract

The sixth-order Boussinesq equation describing Rossby waves in the barotropic atmosphere is derived from the quasi-geostrophic vorticity equation using scale analysis and perturbation expansion. The symmetry and conservation laws of the equation are analyzed. The solution to the equation is obtained through the Jacobi elliptic function expansion method, and the effect of high-order terms and wave numbers on the waves are discussed. The results show that both the higher order terms and the numbers of waves affect the height of the amplitude of the wave. When the high-order terms are present, the height of the amplitude is lower than when the high-order terms are not present and when the wave number is larger, the height of the amplitude is also higher, and this conclusion is not affected by the presence or absence of high-order terms. Finally, the stability of the equation is analyzed by the concept of linear stability analysis. It is pointed out that when the higher order term does not exist, the stable region of the equation disappears and the equation becomes unstable.