The velocity vector field given by
$$ \vec{U}(x, y) = \langle u_x, u_y \rangle = \langle a cos(\lambda x), a sin(\lambda x) \rangle,$$
where a is the fixed amplitude, and lambda is the fixed wavelength, does not describe a circle with radius $a$; instead, plotting this vector field in matlab shows a nice, wavy profile.
Where am I mistaken? To me, it looks like the parameterization of a circle.
If we look at your equation, assuming that your parameter x is in fact the polar angle $\theta$, then our vector field looks like. $$\vec u=a (cos(n \theta), sin(n \theta))$$ The we have a field of vectors, whose strength doesn't depend on position, but the directions rotates as a multiple of the angle.
It's a little bit like the rotation and revolution of a planet. As we look at various angles around the origin, the direction of the vector field changes.
These diagrams look like
$n=1$
$n=2$
$n=3$
$n=4$
If you really do mean x and not the polar angle $\theta$, then our equation is $$\vec u=a (cos(n x), sin(n x))$$
and you get diagrams that look like this.
And this
As we move from left to right along the x direction, the direction of the vector field rotates. Up and down along the y direction the vector field is constant.