A note on acoustic turbulence

We consider a three-dimensional acoustic field of an ideal gas in which all entropy production is confined to weak shocks and show that similar scaling relations hold for such a field as for forced Burgers turbulence, where the shock amplitude scales as $(\epsilon d)^{1/3}$ and the $p$:th order structure function scales as $(\epsilon d)^{p/3} r/d$, $\epsilon$ being the mean energy dissipation per unit mass, $d$ the mean distance between the shocks and $r$ the separation distance. However, for the acoustic field $\epsilon$ should be replaced by $\epsilon + \chi$, where $\chi$ is associated with entropy production due to heat conduction. In particular, the third order longitudinal structure function scales as $\langle \delta u_r^3 \rangle = -C(\epsilon + \chi) r$, where $C$ takes the value $12/5(\gamma +1)$ in the weak shock limit, $\gamma = c_p/c_v$ being the ratio between the specific heats at constant pressure and constant volume.