Unsteady Boussinesq triangular-profile line-plume balances
= Unsteady Boussinesq triangular-profile line-plume balances
In the <Boussinesq approximation>, density-weighted fluxes are $Q=\rho_0bW$, $M=(2/3)\rho_0bW^2$ and $F=(2/3)gbW(\rho_0-\rho)$. With two-sided <Batchelor entrainment> $e=\alpha W$, the integral balances are
$$
\frac43(Q^2/M)_t+Q_z=3\alpha\rho_0M/Q,\qquad Q_t+M_z=FQ/M,\qquad(FQ/M)_t+F_z=0.
$$
The factors $4/3$ and $3$ depend on the triangular profile and the definition of the <entrainment coefficient>.