In my study of fluid dynamics I have across these terms quite often and I am so far unable to create a physical picture in my head. Suppose we have a 3D velocity field $u$, and its corresponding vorticity field is $\omega = \nabla \times u$. What exactly is a vortex line, a vortex tube, a vortex sheet, and a vortex filament? While we are at it, what is the difference between a point vortex and a vortex line?
My current understanding is that a vortex line is simply an integral curve through the vorticity field (i.e. choose a point and "follow" the vector field to see where the point came from and where it is going). If this physical picture is correct, then a vortex tube is defined as a bundle of vortex lines. What is the precise definition of a bundle here? I have seen a vortex sheet as a surface that is everywhere tangent to $\omega$. Is this now simply an integral surface? As for vorticity filaments, I have no idea what that is currently.
I suppose another way of wording my question is that I understand, in most cases, what these things mean mathematically, but physically I cannot create a picture of what they are.