Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 5 / b / Solution

The mass and heat entrainment equations immediately give

\frac{d}{d z} (F - c_{p} T_{0} Q) = 0.

Use the specified relation

ρT = ρ_{0} T_{0} = β

. Then

F - c_{p} T_{0} Q = c_{p} Q (T - T_{0}) = c_{p} T_{0} Q (\frac{ρ _{0}}{ρ} - 1) .

Meanwhile the kinematic buoyancy flux is

B = g \frac{ρ _{0} - ρ}{ρ _{0}} w b^{2} = \frac{g}{ρ _{0}} Q (\frac{ρ _{0}}{ρ} - 1) = \frac{g}{ρ _{0} c _{p} T _{0}} (F - c_{p} T_{0} Q) .

Since

T_{0}

,

ρ_{0}

and

c_{p}

are constant,

d B / d z = 0.

Thus conservation is a consequence of conserved excess heat in this ideal-gas, uniform-ambient model, not merely an assumption that arbitrary non-Boussinesq buoyancy fluxes are conserved in every environment.

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