Dear student and amateur astronomer Haimoud,

The short answer to your precise question is the next: Adaptive optics is used to correct optical images of stars because the recommendations resulting from various theories of turbulence, which include abstract mathematical symbols, do not correspond to the nature of physical phenomena.

The atmosphere is stratified, layered and fine structured. Variations in the refractive index, which, like density, can change sharply in thin layers, cause large deviations of the rays. Turbulence theories calculate averaged smoothed quantities that are absent in the real medium.

More rigorously.

The theory of atmospheric flows is based on the system of fundamental equations of d'Alembert, Navier-Stokes, Fourier and Fick - differential analogues of the of matter, momentum and energy conservation laws, supplemented by the state equation for density [1,2]. Analysis of the invariant properties of the system using methods of the theory of continuous groups shows its compliance with the basic principles of physics [3]. The mathematical quantities included in the system correspond to the observed physical analogues.

The equations of turbulence theory are obtained as a result of applying a number of non-identical transformations of the original fundamental system, supplemented by introducing new hypotheses, functions, constants that change the meaning of the original symbols. Moreover, each theory introduces new quantities that must be designated by new symbols, which is not done "to shorten the description". The symmetries of the equations of turbulence theories do not correspond to the fundamental conservation laws [4]. Accordingly, the solutions of the turbulence equations do not describe the behavior of real physical systems.

In other words, theories of turbulence do not allow calculating real deflections of rays in the atmosphere and cannot be used in real time to correct phase fronts. What do such theories allow? They provide rough estimates of the size of the area into which deflected rays can fall.

To clarify, a next analogy can be given.

In thermodynamics, the concept of "efficiency" introduced by S. Carnot is widely used. Nobody builds heat engines with slow cycles. However, the "efficiency factor" is widely used to compare the relative efficiency of operating engines built on other physical principles.

A more complete stratified flow components classification is given in [3,5].

Examples of visualization of the fine structure of stratified flows in laboratory basins with stratified fluid, produced by analogues of astronomic instruments, are given in [6,7] and other papers, a list of which can be found on the author's page in Research/Gate.

With best regards,

Yuli D .Chashechkin

References.

1. Landau L.D., Lifshitz E.M. Fluid Mechanics. V.6. Course of Theoretical Physics. Oxford: Pergamon Press, 1987.

2. Vallis G.K. Atmospheric and oceanic fluid dynamics. Cambridge: CUP, 2017.

3. Chashechkin Yu.D. Foundations of engineering mathematics applied for fluid flows // Axioms. 2021. V.10. P. 286. https://doi.org/10.3390/axioms10040286.

4. Baidulov, V. G. & Chashechkin, Yu. D. Invariant properties of the equations of motion of stratified fluids // Dokl. Phys. 2002. 47 (12), 888–891.

5. Chashechkin, Y.D., Ochirov, A.A. Periodic flows in a viscous stratified fluid in a homogeneous gravitational field // Mathematics 2023, 11, 4443. https://doi.org/10.3390/math11214443.

6. Chashechkin, Y.D. Discrete and Continuous Symmetries of Stratified Flows Past a Sphere // Symmetry 2022, 14(6), 1278. DOI:10.3390/sym14061278.

7. Chashechkin Yu. D. Schlieren and Electrolytic Visualization of Fluid Flows // Technical Physics, 2024. 2024, Vol. 69, No. 1, pp. 104–107. DOI: 10.1134/S1063784224700373.

Atmospheric turbulence is a 2D Stochastic field of index of refraction fluctuations - there are no predictive numerical methods to be applied.