In fluid dynamics I have come across two sets of equations, the Navier-Stokes equation and the Lorenz equations. From what I have read the Navier-Stokes equations always holds. So why do we need the Lorenz equations also? and when would we use them over the Navier-Stokes equations?
The Navier-Stokes equations are a pretty good description of fluid dynamics, whereas the Lorenz equations are a toy model for the atmosphere.
Like with any toy model, you use the Lorenz equations when you want to model some basic aspect of a real system that you believe is already conveyed by the simplified model. It won't give you realistic results, but might be enough to give you valuable insight into the problem.