Equation 1 is the conservation of momentum and equation 2 is the conservation of energy without taking into consideration turbulence. Equation 3 is basically a turbulence model, which means its a simple way to provide a description of turbulent flow, which is normally very hard to predict. Equation 4 then combines the internal energy with and without turbulence to provide an equation that includes both. I could have made some mistakes there, however all of those equations listed also have citations when they are presented in the paper, I would suggest looking at those cited papers as well for some additional background. In addition, I would suggest some research in fluid dynamics, as well as fluid dynamics applied to astrophysics for to get a better understanding of where those equations come from, and why they are in that specific form.
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