What is the long wavelength(low wave number) limit in hydronamics? And the high-order effect is (short wavelength limit)?

As the name suggests it is the limit where the waves that propagate in the fluid are long compared to some characteristic length scale of the media. In water waves on a layer of finite (mean) depth H then the long wave limit is concerned with waves whose length L=(2.pi/k) is large compared to H; i.e. H/L << 1.

The short wave limit ( not the high order effect which is concerned with correction terms in the perturbation expansion ) has H/L >> 1.