Renormalization-group theory for the propagation of a turbulent burst

We consider the propagation of a plane front separating a turbulent region of fluid from a quiescent region. Initially, the turbulent-energy distribution as a function of z, the displacement normal to the front, is assumed to be localized, and after a time t, general renormalization-group arguments show that there is a similarity solution of the form q(z,t)∼${\mathit{t}}^{\mathrm{-}(2/3+2\mathrm{}\mathrm{\alpha }\mathrm{̃})}$f (${\mathit{zt}}^{\mathrm{-}(2/3+\mathrm{\beta })}$, ε), where α̃ and β are ε-dependent anomalous dimensions, satisfying the scaling law α̃+β=0 and ε is a measure of the dissipation. Using perturbation theory, we calculate values of α̃ and β to O(ε), which are in good agreement with numerical calculations, and we explicitly verify the above scaling law and find the form of the scaling function f.