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This function specifies the fraction of disrupted mass assigned to each fragment size interval. A self-similar law depends on fragment-to-parent size ratios, rather than an absolute size. In logarithmic bins its translation invariance gives the constant processed mass flux of the interior steady collisional cascade. In a removal-dominated interval, it instead sets the fragment source spectrum.

Dividing the mass flux, momentum flux, and density-weighted buoyancy flux by ambient mass density gives the kinematic fluxes $q=\pi b^{2}V$, $m=\pi b^2{V}^2$, and $f=\pi b^{2}Vg({\rho }_0-\rho )/{\rho }_0$. This convention makes dimensional similarity solutions independent of the arbitrary ambient density unit.

For a turbulent plume in a homogeneous ambient of density ${\rho }_0$, define actual sectional mass flux, momentum flux and density-weighted buoyancy flux by $Q=\pi b^2\rho w$, $M=\pi b^2\rho w^2$ and $F=\pi b^{2}wg({\rho }_0-\rho )$. With non-Boussinesq Batchelor entrainment $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, their balances areHere $Q^2/M$ is sectional mass, $QF/M$ is sectional density-deficit buoyancy, and the entrained ambient initially has zero vertical momentum. The last equation follows by subtracting mass conservation from ${\rho }_0$ times volume conservation. Scase, Caulfield, Dalziel and Hunt's time-dependent plume model derives the equivalent system with the common factor $\pi$ suppressed.