Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 345 / 3 / c

Solution

Choose

ρ_{0}

as a representative room density, for example the fresh-air density, and neglect relative density variations everywhere except in buoyancy. With

g^{'} = \frac{ρ - ρ}{ρ _{0}} g,

the triangular profiles give

Q = \frac{1}{2} ρ_{0} bW, M = \frac{1}{3} ρ_{0} b W^{2}, F = \frac{1}{3} ρ_{0} bW g^{'} .

Solving these algebraic relations,

W = \frac{3 M}{2 Q}, b = \frac{4 Q ^{2}}{3 ρ _{0} M}, g^{'} = \frac{3 F}{2 Q} .

\frac{d V}{d z} = α W, \frac{d Q}{d z} = ρ_{0} α W = \frac{3 ρ _{0} α M}{2 Q}, \frac{d M}{d z} = \frac{1}{2} ρ_{0} b g^{'} = \frac{QF}{M} .

An ascending parcel entrains ambient fluid from progressively lower ambient density. With the buoyancy frequency

N^{2} = - \frac{g}{ρ _{0}} \frac{d ρ}{d z},

the change of ambient reference density subtracts

Q N^{2}

from its density-weighted buoyancy flux, so

\frac{d F}{d z} = - Q N^{2} .

There are no discussions about this article yet.

There are currently no matching articles.