Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 4 / d / Solution

The dominant balance is convection of small turbulent energy and dissipation by the steady mean flow against their modeled turbulent diffusion. Diffusion supplies turbulent quantities from the turbulent side to the front; there is no local turbulent production in this problem. For $q=2p-1$, the energy-destruction term divided by its advection term is proportional to ${\xi }^{q-(p-1)}={\xi }^p$. In the dissipation equation, the destruction term has exponent $2q-p$, so its ratio to advection is likewise ${\xi }^{(2q-p)-(q-1)}={\xi }^p$. Both tend to zero for $p>0$. Thus advection balances turbulent diffusion, while both modeled destruction terms are asymptotically smaller.