Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 90 / 4 / b / Solution

Keep the half-section convention of part (a). Define the mass density-weighted mass flux, buoyancy flux and momentum flux byThe full-section fluxes are $2Q,2F,2M$. These factors must be changed consistently if full fluxes are used; mixing a full-width flux with a half-width entrainment source would introduce a spurious factor two.

For positive $Q,M$, $w=M/Q$, $\rho b=Q^2/M$, and $({\rho }_0-\rho )gb=FQ/M$. The mass equation in part (a), the momentum equation, and the buoyancy equation therefore becomeThese are direct algebraic rewritings with the mass density factors retained. In particular, $FQ/M$ is the buoyancy inventory per unit height as well as the integrated buoyancy force. If desired, $\rho$ can be reconstructed from $F/(gQ)={\rho }_0/\rho -1$.

At leading Boussinesq order use a constant reference mass density $R$ in the inertial fluxes, $Q=Rbw$, $M=Rb{w}^2$, and write $g^{\prime }=g({\rho }_0-\rho )/R$ so $F=R{g}^{\prime }bw$. Set ${\rho }_0/\rho =1$ in the stratification source and use $N^2\simeq -g{\rho }_0^{\prime }/R$. The unsteady top-hat line-plume balances are thenEquivalently the kinematic fluxes $q=Q/R=bw$, $m=M/R=b{w}^2$ and $f=F/R=g^{\prime }bw$ obey the same equations with $R$ removed from the first source. This explicit distinction between mass density-weighted and kinematic fluxes fixes the normalization for the power-law solution.