differential geometry - Geometric interpretation of $vec v cdot operatorname{curl} vec v = 0$

In this Math.SE question, I asked a question to which I was hoping to get a simple intuitive answer. Instead I received an otherwise perfectly correct but very mathematical one. Obviously, the words geometric and intuitive have very different meaning in that forum from what they mean to me... Can somebody help here?

There is a family of surfaces orthogonal to the vector field $\vec v \in \mathbb R^3$ iff $\vec v \cdot \operatorname{curl} \vec v = 0 $. Now the necessity part is trivial, but the proof of sufficiency I have seen in physics textbooks, e.g. Kestin: Thermodynamics, is kind of mindless integration similar to the one usually offered to prove Poincare's lemma as in Flanders, or just asserted as in Born&Wolf: Optics. Is there an intuitive and geometric interpretation of this condition that would make it obvious why its sufficiency must be true?

Answer

Edit: Now that I've read the original post more closely, I'm not sure that this answer actually addresses your question. Let me know if it doesn't and I can perhaps try again.

Remember: curl is only defined for vector fields and not for lone, individual vectors. With this in mind, curl is sometimes said to quantify the "vorticity" of a vector field (i.e. how swirly it is at a given point). Intuitively, if you imagine putting a little, tiny paddle-wheel into the vector field and think of the field as a flowing stream and the magnitudes of the nearby vectors as the strength of the current at that point, then you can tell whether the curl of the field is zero at the point described by the center of the paddle-wheel by figuring out whether or not the paddle-wheel is caused to turn by the current. Moreover, the faster the paddle-wheel turns the greater the vorticity and, thus, the greater the curl.

Now a spinning wheel lives in a plane, yes? That is to say, uniform rotation about a point is a fundamentally two-dimensional activity. Once we've observed this fact, we can easily see that the best way to define a plane is to give the components of its normal vector. Since the plane is flat and uniform out to infinity, the normal vector does not change and, more importantly, the normal vector is perpendicular to the plane at all points.

So now we see that the curl is giving us the information of a plane which we encode and compactify into a single vector. We do this again at every point in space and we get a new vector field that is the curl of the original vector field. This new vector vield has the interesting property that at any given point, the vector there is perpendicular to the vector at that point in the original field.

And the dot product of perpendicular vectors is always zero. Intuitive enough for you?

So to summarize, the curl of a vector field at a point gives you a vector that is normal to the plane of rotation of the original vector field. The magnitude of the new vector tells you how curly it is, while the direction tells you which way the original field is spinning. By the way, don't forget the right-hand rule.