The Euler equation (EE) is one of the basic equations in many physical fields such as fluids, plasmas, condensed matter, astrophysics, oceanic and atmospheric dynamics.A new symmetry group theorem of the (2+1)-dimensional EE is obtained via a simple direct method which is thus utilized to find exact analytical vortex and circum fluence solutions.A weak Darboux transformation theorem of the (2+1)-dimensional EE can be obtained for arbitrary spectral parameter from the general symmetry group the orem.The possible application of the vortex and circumfluence solutions to the tropical cyclones, especially the Hurricane Katrina 2005, is demonstrated.Tsunami waves usually are described by the exact solutions of the (1+1)-dimensional dispersionless water wave equations (Science 309, 2045 (2005); Phys.Rev.Lett.97 (2006) 148501), the soliton solutions of the integrable models like KdV and KP equations (arXiv:nlin.PS/0702052 (2007)), In this report, we discuss also the possibility to describe the Tsunami waves by folded soliton or folded solitary waves (X.Y.Tang and S.Y.Lou, J.Math.Phys.44 (2003) 4000).