fluid dynamics - Waves: Small Amplitude Approximation

Why does the assumption of waves amplitude being small, compared to the wavelength, (as used for surface gravity waves in water) lead to the equivalent condition: $$ {\partial v \over \partial t } >> { (v \nabla )v}$$

This seems to be one of the basic assumptions of certain classes of fluid mechanic problems.

I spend two hours in the library today reading the applicable chapters of about nine books and not in one the explanation was more detailed that to claim that $ (v \nabla )v$ must be on the order of $v^2<<1$.