Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 4 / b

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- Solution b

Solution

With thermal diffusivity

κ_{T}

, temperature obeys

T_{t} + u T_{x} + w T_{z} = κ_{T} (T_{xx} + T_{zz}) .

Scale

z

with

h_{0}

,

x

with

b_{0} = ϵ h_{0}

, and

w

with

w_{0}

. Continuity gives the cross-plume velocity scale

u_{0} \sim (b_{0} / h_{0}) w_{0} = ϵ w_{0}

, so both advective terms scale as

w_{0} Δ T / h_{0}

. Horizontal and vertical diffusion scale as

κ_{T} Δ T / b_{0}^{2}

and

κ_{T} Δ T / h_{0}^{2}

, respectively. Their ratio is

\frac{vertical diffusion}{horizontal diffusion} = \frac{b _{0}^{2}}{h _{0}^{2}} = ϵ^{2} ≪ 1 .

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