Referee #1 (Remarks to the Author):
In this nice paper, the author prove the convergence rate of compressible Euler and Rotating Shallow Water equations in a singular limit process when Mach or Rossby number tends to zero. The problem is fundamental in fluid dynamics, while it is very difficult when boundary presented and the initial data is not well-prepared. The problem becomes even harder when the boundary is characteristic. These all happen in current paper. The author managed to overcome these difficulties through an effective elliptic estimates and some nice ideas. I thus recommend to accept this paper for publication in SIMA. The paper is very well-written. There are only couple isolated typos. Therefore, the author should read the paper carefully and make correction accordingly.