"Height field" is mathematical model of land.

Electromagnetic filed is mathematical model of what?

Electromagnetism is a U(1) gauge theory. If you take the accompanying classical geometry at face value, there's hidden state at each point in space, like some sort of dial. The absolute position of the dial is irrelevant, but the gauge potential tells you how the dial turns as you move from place to place. The electromagnetic field strength is something called curvature of the corresponding 'connection'. The mathematics are a bit involved, but morally speaking, I would say it tells you how the turning of the dial varies across space.

So, EM field describes strength of EM force in every point of mathematical model. Nature of EM force is unknown. Right?

Let's talk about nature of EM force.

Analogy: look at a typical tropical cyclone. It rotates. Is it rotating because a unknown property of an air molecule? No. It rotates because our planet rotates, while air molecules are just trying to keep their positions. I.e. it's rotation of planet + inertia of molecules.

Is it possible that EM force is happens because our local space is moving trough global space by non-linear trajectory, so it just non-linear trajectory of local space + inertia of rotating and vibrating particles?

Not exactly only mathematical, though - you can feel it as you walk across.

I feel also air field (wind), height field (land), light field (photons), and so on.

My point exactly - fields are real, they are not merely mathematical models. (It's just that some words acquire a more precise meaning, being formalized as part of a model, and while it's true that models may contain an additional "scaffolding" that has no analogue in reality, field is not one of those.)

Field is mathematical abstraction, used in mathematical model, to represent a physical thing.

Physical things are real. Mathematical abstractions are not. OpenGL is not real too, but it looks very real and accurately predicts reality. In OpenGL, field is array, e.g. "float[][][] field;".

That’s like saying, “Cat is a word used in a language to represent a physical thing.”

No, it's like saying that nature is built on kind of vertex buffer, because OpenGL accurately predicts reality.