The use of differential equations to describe nature dates back to 1677 and the fundamental anagram of calculus found in Newton's letter to Leibniz:

The secret form of this anagram consist of the following

6a cc d ae 13e ff 7i 3l 9n 4o 4q rr 4s 8t 12u x

After decoding the anagram, we will get:

Data (ae)quatione quotcunque fluentes quantitates involvente, fluxiones invenire; e(t) vice versa

ae - single character, t in et not counted

The prototype of the calculus was formulated in terms of currents and flexions but for the first time we see the idea of combining functions and derivatives in a single equation.

The next step was taken by Leibniz and later by Euler who formulated the equations describing fluid flow.

The succinct summary of these seminal contributions was formulated by Clifford Trusdell

“The theories of fluid dynamics, acoustics, thermal conduction, elasticity, thermodynamics, electricity and magnetism, and relativity are successive triumphs of the direct Eulerian method, and even to describe strictly molecular phenomena, the Hamilton – Jacobi theory, the Gibs mechanics, and quantum mechanics have recourse to indirect and more elaborate field descriptions”

Clifford Trusdell (1954) The kinematics of vorticity

The basic equation describing the dynamics is the Navier-Stokes equation which is an extension of Newton's 2nd law to fluid motion (conservation of momentum), plus equations describing the thermodynamics (conservation of energy), plus a continuity equation (conservation of mass/matter).

Please consider also the following article introducing the historical background of Numerical Weather Prediction

https://maths.ucd.ie/~plynch/Publications/JCP.pdf

Thank you Prof. Pudykiewicz and Prof. Mestas-Nunez for sharing your knowledge. I think more research, more discussion and time ( knowledge generation from time series analysis) will tell the actual answer. Thanks.