  We prove global existence and asymptotic behavior of classical solutions for
two dimensional inviscid Rotating Shallow Water system with small initial data
subject to the zero-relative-vorticity constraint. One of the key steps is a
reformulation of the problem into a symmetric quasilinear Klein-Gordon system,
for which the global existence of classical solutions is then proved with
combination of the vector field approach and the normal forms. We also probe
the case of general initial data and reveal a lower bound for the lifespan that
is almost inversely proportional to the size of the initial relative vorticity.
