Is studying turbulence from within mathematics or physics going to be useful when studying leadership, or is it just a metaphor? Or are turbulence and leadership what is known as a category error?
If I understand your question properly, turbulence is physical phenomenon. However, in many cases turbulence is nothing more than metaphor. Say, somebody have no understanding on the phenomenon that he is studying. In this case hi is saying oh it is turbulence. Sometimes (erroneously) turbulence is associated with ergodicity. It is not the case. However, usually when studying the problem analytically some properties are neglected. Successful following this line is Kolmogorov solution.