Existence of closed level sets on a surface for some field

Question

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. At each point of this surface, we can record the wind speed having some magnitude (say like Einstein puts tiny clocks at all coordinates of space to measure time). If at any time instant $t$, this undescribed mechanism records all values of speeds over the surface without repetition as follows

$\mathscr{V}=[v_1,v_2,...,v_n]$

1. Then for some $v\in \mathscr{V}$, can we draw level set-type plots on the surface by joining all points having same speed $v$ by a curve on the surface such that:-
   a) if such curves exist, they necessarily form closed loops?
   b) And there are non-zero number (meaning at least one or more) of closed contours on the surface?

2. What are the seminal sources where existence proofs of such kind are studied that analyse the necessary and sufficient conditions for existence of closed level sets for a given field?

3. What are the restrictions (if any) to be imposed on the (a) geometry of surface and (b) wind flow, to ensure the invariable existence for all time of such closed level sets on the surface?

Constructive expositions/proofs will be better.