Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 345 / 3 / a

Take inward fluid entrainment speed to be positive. The non-Boussinesq form of the Batchelor entrainment hypothesis iswhere $\alpha$ is the entrainment coefficient. Define physical, cross-section-integrated fluxes for the top-hat plume model byThus $Q$ is mass flux, $M$ is momentum flux, and $F$ is a density-weighted buoyancy flux. The kinematic buoyancy flux in the Boussinesq approximation is $F/{\rho }_0$. These definitions keep the factor $\pi$; conventions suppressing it simply rescale all three fluxes consistently.

For a homogeneous ambient, the sectional volume, mass and momentum balances areThe momentum source is the net upward buoyancy force; entrained ambient fluid initially supplies no vertical momentum. Subtracting the mass balance from ${\rho }_0$ times the volume balance gives conservation of the density deficit:Now $w=M/Q$, $\pi b^2\rho =Q^2/M$, and $\pi b^{2}g({\rho }_0-\rho )=QF/M$. Substitution produces the non-Boussinesq top-hat plume equationsFor reconstructing the fields without a Boussinesq approximation,The last expression uses the plume density in the acceleration denominator; it agrees with the ambient-density definition of reduced gravity to Boussinesq order.