Existence of closed level sets on a surface for some field 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
$\mathscr{V}=[v_1,v_2,...,v_n]$


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*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?  

*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?

*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.
 A: To address the question as clarified in the comments: Generally, you're asking questions of differential topology. The book of that title by Victor Guillemin and Alan Pollack is a clear, detailed introduction to the ideas sketched here. Morse Theory by John Milnor may also be of interest.
tl; dr: The following tools are useful for studying level sets of smooth functions and flows of smooth vector fields:


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*The Implicit Function Theorem;

*The Hopf Index Theorem;

*Morse Theory.

The simplest setting for each is a compact, smooth manifold $M$ without boundary, equipped with a smooth vector field $X$ having isolated zeros, or equipped with a "generic" smooth function $f$. The technical definition of "generic" is a bit fussy: $f$ has isolated critical points, the Hessian of $f$ is non-singular at each critical point ("non-degenerate critical points"), and each level set of $f$ contains at most one critical point ("distinct critical values"). A function satisfying all three conditions is called a Morse function.
In your situation, if $f$ is a Morse function on the sphere $S^{2}$, then:


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*The Implicit Function Theorem implies that each non-empty level set of $f$ is a single point; a smooth, closed loop; a smooth figure eight with two distinct tangent lines at the point of crossing; or a finite disjoint union of such sets, with at most one component not a smooth loop.


(Sketch of proof: A non-empty level set containing no critical points is a smooth, compact curve, hence a finite union of loops. Otherwise, the level set contains a single critical point $p$. If $p$ is a local maximum or minimum of $f$, the one-point set $\{p\}$ is a component of the level set of $f$; if $p$ is a non-degenerate saddle point, the component of the level of $f$ through $p$ is a figure eight with $p$ as a crossing.
If you continue to require isolated, non-degenerate critical points but allow level sets of $f$ to contain more than one critical point, then a level of $f$ comprises a finite union of: isolated points; smooth, closed loops; figure eights as above (a.k.a., immersed circles with one crossing); immersed circles with multiple (but finitely many) crossings.
If you allow $f$ to have isolated, degenerate critical points (i.e., the Hessian can have determinant zero), the level curves can be substantially more complicated, including points where an even number of rays meet.
If in addition you allow $f$ to have non-isolated critical points, then an arbitrary closed subset of the sphere is a level set of some smooth function $f$.)


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*Let $f$ be a Morse function. Define the (Morse) index of a local minimum to be $0$, of a saddle point to be $1$, and of a local maximum to be $2$. The alternating sum of the indices of the critical points of $f$ is equal to the Euler characteristic of the sphere. (An analgous claim holds on an arbitrary smooth, compact manifold without boundary.)

*Let $X$ be a smooth (tangent) vector field on the sphere with isolated zeros. At each zero $p$ of $X$, define the (Hopf) index to be "the number of times the value of $X$ rotates along a small circle about $p$". (The Wikipedia page gives a more formal definition.) The sum of the indices (taken over all zeros of $X$) is the Euler characteristic of the sphere. (An analgous claim holds on an arbitrary smooth, compact manifold without boundary.)
Morse Theory and the Hopf Index Theorem are not unrelated: If $f$ is a smooth, real-valued function on the sphere with isolated, non-degenerate critical points, its gradient $X = \nabla f$ is a smooth vector field with isolated zeros, and the sign of the determinant of the Hessian of $f$ at a critical point $p$ is the Hopf index of $X$ at $p$. 
For example, the function $f(x, y, z) = x^{2} - z^{2}$ on the unit sphere (which has isolated, non-degenerate critical points, but does not have distinct critical values) has the following levels $\{f = c\}$:


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*$c = -1$: Two local minima, $(0, 0, \pm 1)$;

*$c = 1$: Two local maxima, $(\pm 1, 0, 0)$;

*$c = 0$: Two circles (the intersection of the sphere with the planes $x = \pm z$) crossing at the saddle points $(0, \pm 1, 0)$;

*$0 < |c| < 1$: Two smooth, non-planar loops (the intersection of the sphere with the hyperbolic cylinder $x^{2} - z^{2} = c$).
The alternating sum of the Morse indices is $2 \cdot 0 - 2 \cdot 1 + 2 \cdot 2 = -2 + 4 = 2 = \chi_{S^{2}}$.
