Consider a knife on its back, so its sharp edge points in the upward, z direction. The edge (idealized as extending infinitely far in the y direction) is hot, and by heating adjacent fluid, it creates a rising thermal plume. Introduce a temperature deficit △T (z) that measures the typical difference in temperature between the plume and the surrounding, ambient fluid at height z above the knife edge, and let δ_p(z) be the width of the plume at height z.

(a) Show that energy conservation implies the constancy of δ_p△T v̅_z, where v̅_z(z) is the plume’s mean vertical speed at height z.

(b) Make an estimate of the buoyancy acceleration, and use it to estimate v̅_z.

(c) Use Eq. (18.19) to relate the width of the plume to the speed. Hence, show that the width of the plume scales as δ_p ∝ z^2/5 and the temperature deficit as △T ∝ z^−3/5.

(d) Repeat this exercise for a 3-dimensional plume above a hot spot.

dr dt =xV Boussinesq (3). (18.19)