Even in the most chaotic, turbulent flow, long-lived coherent vortices can be identified (e.g., Figure $$\PageIndex{1}$$). In this section we will establish three theorems that tell us why vortices have such a remarkable tendency to stay together. To begin with, we will study the basic aspects of vorticity in the simplest possible form, by neglecting complications due to viscosity and inhomogeneity, i.e., we will assume that $$\mu=v=0$$ and $$\rho=\rho_0$$. Later, we will add the effects of viscosity and allow $$\rho$$ to vary.