Similarity solution

A similarity solution combines independent variables into scale-invariant coordinates and reduces a PDE to an ODE.

A similarity profile is the time-independent function left after expressing a evolving field in its scaled coordinate, for example $\rho (z,t)={\rho }_{0}f(z/Z(t))$. Integrating such profiles converts a global conservation law into an algebraic relation among the time-dependent scales.