What are the Lorenz Equations used for?

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.

The Lorenz equations are often used to think about Navier-Stokes dynamics from the perspective of "phase space". To do this, parameters such as distance and time from the Navier-Stokes equations must be de-dimensionalized and normalized such that coefficients in the Lorenz equations are dimensionless and only range from [0,1].

In order to do this, LOTS of assumptions about fluid flows need to be made.

To name only a few:

• Fluid flows generate no internal heat (by way of internal friction)
• Fluid behaves in a Newtonian way
• Fluid conduction obeys Fourier's law (roughly meaning that the thermal flow obeys Netwon's second law, is smooth and doesn't have weird disjunctive thermal boundaries across the body)
• ...other stuff

Once all that magic happens, you get static and variable coefficients like the Reynolds number, where the idealized dynamics of fluid flow can be examined as a set of state-based relations.

Confusingly, and contrary to what you might infer from looking at the Lorenz equations on Wikipedia, the Lorenz equations don't have to exist in only a single time-based form. Note the transformation from Navier-Stokes to Lorenz is a transform from a PDE to an ODE.

I found this link to be very helpful and concise at 'splaining the connection between the two equations: http://www.ub.edu/simba/slides/simba10102018JuanPello.pdf